D Hu

T Lelièvre
email: lelievre@cermics.enpc.fr

New entropy estimates for the Oldroyd-B model, and related models

This short note presents the derivation of a new a priori estimate for the Oldroyd-B model. Such an estimate may provide useful information when investigating the long-time behaviour of macro-macro models, and the stability of numerical schemes. We show how this estimate can be used as a guideline to derive new estimates for other macroscopic models, like the FENE-P model.

Introduction

We consider the Oldroyd-B model:

Re ∂u ∂t + u.∇u = (1 -ε)∆u -∇p + div τ , (1)
div (u) = 0, (2) ∂τ ∂t + u.∇τ = ∇uτ + τ (∇u

) T - 1 We τ + ε We ∇u + (∇u) T , (3)
where the Reynolds number Re > 0, the Weissenberg number We > 0 and ε ∈ (0, 1) are some non-dimensional numbers. We suppose that the space variable x lives in a bounded domain D of IR d . This system is supplied with initial conditions on the velocity u and on the stress tensor τ . For simplicity, we assume no-slip boundary conditions on the velocity u: u = 0 on ∂D.

We suppose that the initial data and the geometry are such that there exists a unique regular solution to (1)-( 3) and our aim is to derive some a priori estimates on this solution.

Let us introduce the so-called conformation tensor A = We ε τ + Id. The partial differential equation (PDE) on τ translates into the following PDE on A:

∂A ∂t + u.∇A = ∇uA + A(∇u) T - 1 We A + 1 We Id. ( 5
)
One can check that if

A(t = 0) = We ε τ (t = 0) + Id is a positive definite symmetric matrix, (6)
then this property is propagated forward in time by [START_REF] Jourdain | Long-time asymptotics of a multiscale model for polymeric fluid flows[END_REF] (and, in particular, τ is symmetric). Assuming uniqueness of solution, this can be proven for example by using the probabilistic interpretation of A as a covariance matrix, as explained in Section 3. We will assume throughout this note that ( 6) is satisfied. Concerning the importance of positive-definiteness of A, we refer for example to [START_REF] Keunings | Fundamentals of Computer Modeling for Polymer Processing, chapter Simulation of viscoelastic fluid flow[END_REF]Section 9.8.10] and also to the recent work [START_REF] Fattal | Constitutive laws for the matrix-logarithm of the conformation tensor[END_REF][START_REF] Fattal | Time-dependent simulation of viscoelastic flows at high Weissenberg number using the log-conformation representation[END_REF].

In Section 2, we recall how the classical a priori estimate for the Oldroyd-B model is derived. Next we show how it can be used to derive some bounds on the stress tensor, provided the initial condition satisfies det A(t = 0) > 1. In Section 3, we establish a new estimate, which comes from an entropy estimate on the micro-macro model associated with the Oldroyd-B model (see [START_REF] Jourdain | Long-time asymptotics of a multiscale model for polymeric fluid flows[END_REF]). This estimate provides bounds on (u, τ ) without any assumption on τ (t = 0) (apart from [START_REF] Jourdain | Convergence of a stochastic particle approximation of the stress tensor for the FENE-P model[END_REF]). This new estimate could be useful to study the longtime behaviour of some macro-macro models, or to analyze the stability of some numerical schemes. Current research is directed towards clarifying this.

The classical estimate

Let us first introduce the kinetic energy:

E(t) = 1 2 D |u| 2 . ( 7
)
We easily obtain from (1)-( 2):

Re

dE dt = -(1 -ε) D |∇u| 2 - D τ : ∇u, (8)
where for two matrices A and B, we denote A : B = A i,j B i,j = tr (A T B). On the other hand, taking the trace of the PDE (3) on τ and integrating over D, we get:

d dt D tr τ = 2 D ∇u : τ - 1 We D tr τ .
We thus obtain the following estimate:

d dt Re 2 D |u| 2 + 1 2 D tr τ +(1 -ε) D |∇u| 2 + 1 2We D tr τ = 0. ( 9
)
Remark 1 In terms of A, the energy estimate (9) writes:

d dt Re 2 D |u| 2 + ε 2We D tr A +(1 -ε) D |∇u| 2 + ε 2We 2 D tr (A -Id) = 0. (10)
In Lemma 1 below, we prove that tr τ is positive if det A(t = 0) > 1. This result combined with the estimate (9) thus yields some a priori bounds on (u, τ ) provided det(A)(t = 0) > 1. In particular, it shows that u and τ go exponentially fast to 0 in the long time limit, using [START_REF] Peterlin | Hydrodynamics of macromolecules in a velocity field with longitudinal gradient[END_REF] and the Poincaré inequality:

D |u| 2 ≤ C D |∇u| 2 .
Lemma 1 Let us assume that det A(t = 0) > 1. Then, we have ∀t ≥ 0, det A(t) > 1 and this implies that tr τ (t) > 0.

Proof: Using (5) and the Jacobi identity (which states that for any invertible matrix M depending smoothly on a parameter t, d dt ln det M = tr M -1 dM dt ), we have:

∂ ln(det A) ∂t + u.∇ ln(det A) = 1 We tr A -1 -Id . (11)
Since for any symmetric positive matrix M of size d × d,

(det M ) 1/d ≤ (1/d)tr M, (12)
we obtain

∂ ln(det A) ∂t + u.∇ ln(det A) ≥ d We (det A) -1/d -1 , which we can rewrite in terms of y = (det A) 1/d : We ∂y ∂t + u.∇y ≥ (1 -y) . (13)
This shows that y > 1 if y(t = 0) > 1, and thus that det

A > 1 if det A(t = 0) > 1.
Indeed, using the characteristic method (by integrating the vector field u(t, x)), one can rewrite (13) as

We Dy Dt ≥ (1 -y) .
Now, if y does not remain greater than 1, consider the first time t 0 such that y(t 0 ) = 1.

We have on the one hand Dy Dt (t 0 ) < 0 and, on the other hand (1 -y(t 0 )) = 0. We reach a contradiction.

We thus have det A > 1 and therefore, using again (12), tr

A > d. Since τ = ε We (A -Id), this is equivalent to tr τ > 0. ♦ Remark 2 If det A(t = 0) < 1 (which is the case if tr τ (t = 0) < 0), Equation (13)
shows that det A grows along the characteristics as long as det A < 1.

Entropy estimate

We now consider a micro-macro (or multiscale) formulation of the Oldroyd-B model and some estimates based on entropy, inspired from [START_REF] Jourdain | Long-time asymptotics of a multiscale model for polymeric fluid flows[END_REF].

General derivation of the entropy estimate for micro-macro models

We consider the following system:

                           Re ∂u ∂t (t, x) + u(t, x).∇u(t, x) = (1 -ε)∆u(t, x) -∇p(t, x) + div τ (t, x), div (u(t, x)) = 0, τ (t, x) = ε We I R d (X ⊗ ∇Π(X))ψ(t, x, X) dX -Id , ∂ψ ∂t (t, x, X) + u(t, x).∇ x ψ(t, x, X) = -div X ∇ x u(t, x)X - 1 2We ∇Π(X) ψ(t, x, X) + 1 2We ∆ X ψ(t, x, X).
(14) This system is supplied with initial conditions on the velocity u and on the distribution ψ. We recall that we suppose no-slip boundary conditions (4) on the velocity u. This system corresponds to a micro-macro model of polymeric fluids, the polymer being modelled by two beads linked by a spring with potential energy Π. The configurational variable X ∈ IR d models the end-to-end vector of the polymer. For more details on the modelling, we refer to [START_REF] Bird | Dynamics of polymeric liquids[END_REF][START_REF] Öttinger | Stochastic Processes in Polymeric Fluids[END_REF].

Notice that we could rewrite the former system as a system coupling a PDE and a stochastic differential equation (SDE), replacing the last two equations by:

τ (t, x) = ε We IE (X t (x) ⊗ ∇Π(X t (x))) -Id , ( 15
)
dX t (x) + u(t, x).∇ x X t (x) dt = ∇ x u(t, x)X t (x) - 1 2We ∇Π(X t (x)) dt + 1 √ We dW t . (16)
There, IE denotes the expectation, W t denotes a d-dimensional standard Brownian motion independent from the initial condition (X 0 (x)) x∈D which is such that, ∀x ∈ D, the law of X 0 (x) is ψ(0, x, X) dX.

Let us introduce the kinetic energy:

E(t) = 1 2 D |u| 2 . ( 17
)
We easily obtain:

Re dE dt = -(1 -ε) D |∇u| 2 - ε We D I R d (X ⊗ ∇Π(X)) : ∇u ψ. ( 18
)
We now introduce the entropy of the system, namely:

H(t) = D I R d ψ(t, x, X) ln ψ(t, x, X) ψ ∞ (X) , (19)
= D I R d Π(X)ψ(t, x, X) + D I R d ψ(t, x, X) ln(ψ(t, x, X)) + C, with ψ ∞ (X) = exp(-Π(X)) I R d exp(-Π(X)) , (20)
and C = ln( I R d exp(-Π(X)))|D|. The function H is actually the relative entropy of ψ with respect to the equilibrium distribution ψ ∞ . After some computations (see [START_REF] Jourdain | Long-time asymptotics of a multiscale model for polymeric fluid flows[END_REF]), we obtain:

dH dt = - 1 2We D I R d ψ ∇ ln ψ ψ ∞ 2 + D I R d (X ⊗ ∇Π(X)) : ∇u ψ. (21)
Therefore, introducing the free energy

F (t) = E(t) + ε
We H(t) of the system, we have:

d dt Re 2 D |u| 2 + ε We D I R d ψ ln ψ ψ ∞ +(1 -ε) D |∇u| 2 + ε 2We 2 D I R d ψ ∇ ln ψ ψ ∞ 2 = 0. ( 22
)
Using a logarithmic Sobolev inequality with respect to ψ ∞ and a Poincaré inequality for u ∈ H 1 0 (D), one can then obtain exponential convergence to equilibrium lim t→∞ (u, ψ) = (0, ψ ∞ ) (see [START_REF] Jourdain | Long-time asymptotics of a multiscale model for polymeric fluid flows[END_REF]). For some generalizations to the case u = 0 on ∂D, we refer to [START_REF] Jourdain | Long-time asymptotics of a multiscale model for polymeric fluid flows[END_REF].

The Oldroyd-B case

Let us consider the Hookean dumbbell model, for which the potential Π of the entropic force is:

Π(X) = ||X|| 2 2 . ( 23
)
By Itô's calculus, it is easy to derive from (16) that A = IE(X t ⊗ X t ) satisfies the following PDE:

∂A ∂t + u.∇A = ∇uA + A(∇u) T - 1 We A + 1 We Id. ( 24
)
This translates into the following PDE for τ = ε We (A -Id):

∂τ ∂t + u.∇τ = ∇uτ + τ (∇u) T - 1 We τ + ε We ∇u + (∇u) T . (25)
The Hookean dumbbell model is thus equivalent to the Oldroyd-B model (at least for regular enough solutions). If ψ(0, x, .) is Gaussian (with zero mean), so is ψ(t, x, .):

ψ(t, x, X) = 1 (2π) d/2 det(A) exp - X T A -1 X 2 where A = IE(X t ⊗ X t ) = I R d X ⊗ X ψ(t,
x, X) dX denotes as above the covariance matrix of X t , which depends on time and also on the space variable x. The covariance matrix A is symmetric and nonnegative. Moreover, since for almost all t ≥ 0, D I R d ψ(t, x, X) ln ψ(t,x,X) ψ∞(X)

< ∞, then for almost all t ≥ 0 and for almost all x ∈ D, A is positive.

The following explicit expression of the relative entropy can then be derived:

D I R d ψ(t, x, X) ln ψ(t, x, X) ψ ∞ (X) dX = D 1 2 (-ln(det A) -d + tr A) .
On the other hand,

D I R d ψ(t, x, X) ∇ X ln ψ(t, x, X) ψ ∞ (X) 2 dX = D tr ((Id -A -1 ) 2 A).
Rewriting (22), we thus obtain the following estimate, in terms of A:

d dt Re 2 D |u| 2 + ε 2We D (-ln(det A) -d + tr A) +(1 -ε) D |∇u| 2 + ε 2We 2 D tr ((Id -A -1 ) 2 A) = 0.
(26) This is, in the specific case of Hookean dumbbells (that is Oldroyd-B model) the macroscopic version of (22).

Since -ln(det(A)) -d + tr (A) ≥ 0, this energy estimate yields some a priori bounds on (u, A), and thus on (u, τ ). In sharp contrast to the classical estimate [START_REF] Peterlin | Hydrodynamics of macromolecules in a velocity field with longitudinal gradient[END_REF], it provides bounds on (u, τ ) without any assumption on τ (t = 0) (apart from ( 6)). Using a Poincaré inequality and the fact1 that, for any symmetric positive matrix M of size d × d, -ln(det M ) -d + tr M ≤ tr ((Id -M -1 ) 2 M ) exponential convergence to equilibrium (lim t→∞ (u, A) = (0, Id)) can be obtained from (26).

Remark 3 Notice that (26) can be schematically obtained as (10)-ε 2We D (11).

Remark 4 If ψ(0, x, .) is not Gaussian, it is always possible to replace it by a Gaussian initial condition with the same mean and variance, so that the macroscopic quantities (u, p, A) would be the same for the two initial conditions.

Application to related macroscopic models

The energy estimate (26) can be used as a guideline to derive energy estimates for other macroscopic models, even though they cannot be recast as a microscopic model of the form (14). Let us consider the example of the FENE-P model [START_REF] Peterlin | Hydrodynamics of macromolecules in a velocity field with longitudinal gradient[END_REF][START_REF] Bird | Polymer solution rheology based on a finitely extensible bead-spring chain model[END_REF], for which

τ = ε We A 1 -tr (A)/b -Id , (27)
∂A ∂t + u.∇A = ∇uA + A(∇u) T - 1 We A 1 -tr (A)/b + 1 We Id. (28)
For this model, we assume [START_REF] Jourdain | Convergence of a stochastic particle approximation of the stress tensor for the FENE-P model[END_REF], and also that tr (A)(t = 0) < b, and this property is propagated forward in time by (28) (see [START_REF] Jourdain | Convergence of a stochastic particle approximation of the stress tensor for the FENE-P model[END_REF]).

Using the same ideas as for the Oldroyd-B model, we consider the "entropy" H(t) = -ln(det A) -b ln (1 -tr (A)/b), and we compute its time-derivative:

d dt D -b ln (1 -tr (A)/b) = 2 D ∇u : A 1 -tr (A)/b + 1 We D - tr (A) (1 -tr (A)/b) 2 + d 1 -tr (A)/b , (29)
d dt D ln(det(A)) = 1 We D - d 1 -tr (A)/b + tr (A -1 ) . (30)
Combining these expressions with (8), we obtain

The proof of these inequalities is tedious and can be done by diagonalizing the matrix M . Equation (32) shows that Equation (34) (which plays the role of the log-Sobolev inequality in the micromacro models) shows that the estimate (31) can be used to prove exponential convergence to equilibrium.

2 + ε 2We 2 D

 22 (det A) -b ln (1 -tr (A)/b)) +(1 -ε) D |∇u| tr (A) (1 -tr (A)/b) 2 -2d 1 -tr (A)/b + tr (A -1 ) = 0.(31) One can check that for any symmetric positive matrix M of size d × d:-ln(det(M )) -b ln (1 -tr (M )/b) ≥ -(b + d) ln b b + d ≥ d(32)and that -ln(det(M )) -b ln (1 -tr (M )/b) + (b + d) ln b b + d (33) ≤ tr (M ) (1 -tr (M )/b) 2 -2d 1 -tr (M )/b + tr (M -1 ) .

Re 2 D

 2 |u| 2 + ε 2We D -ln(det A) -b ln (1 -tr (A)/b) + (b + d) ln b b + dis a non-negative quantity, and thus that (31) indeed yields some a priori bounds on (u, A).

which can be seen as the logarithmic Sobolev inequality for Gaussian random variables translated on their covariance matrices


Martin Heusse

Timothy X Brown

Thomas Schwengler

Andrzej Duda

A new look at bidirectional TCP connections over asymmetric links

Many papers invoke the ACK compression phenomenon to explain the download performance drop in the presence of bidirectional traffic, whereas we argue that this symptom is merely a side effect of the core phenomenon that we analyze here. Interferences between upload and download is more and more frequent as it requires two commonplace ingredients: asymmetric links like most SOHO access links and long-lasting TCP uploads e.g. due to the presence of peer-to-peer traffic.

We find that interaction between uploads and downloads is simple to analyze in the rare case where the (up)link buffer size is given in bytes. Otherwise, in the more common case where buffer size is a given number of packets, there is greater adverse interactions. We provide guidance for sizing the link buffers to mitigate detrimental interference. We provide experimental results that illustrate the case with buffer sizes in bytes using an asymmetric WiMAX link.

L e s r a p p o r t s d e r e c h e r c h e d u L I G Ma r t i n H E U S S E , P r o f e s s o r , U . J o s e p h F o u r i e r , G r e n o b l e , F r a n c e T i mo t h y X B R O WN , A s s o c i a t e d P r o f e s s o r , U . o f C o l o r a d o , B o u l d e r , U S A T h o ma s S C H WE N G L E R , I n s t r u c t o r A d j u n c t , , U . o f C o l o r a d o , B o u l d e r , U S A A n d r z e j D U D A , P r o f e s s o r , G r e n o b l e -I N P , G r e n o b l e , F r a n c e

A n e w l o o k a t b i d i r e c t i o n a l T C P c o n n e c t i o n s o v e r a s y mme t r i c l i n k s R R -L I G -0 2 ma r s 2 1 0

Introduction

We study the interactions between upload and download traffic on asymmetric links. This phenomenon, which often causes drastic throughput reduction of the TCP downloads, is not well understood although this type of link is pervasive at the edge of the Internet. More precisely, in many cases, data transfers in each direction obtain similarly low throughputs, which on a typical ADSL line leads to filling only one tenth of the downlink capacity in the presence of uploads. This problem is rendered more acute by the evolution in network usage as users tend to upload more and more data, often continuously in the case of P2P software [START_REF] Schulze | Internet study[END_REF]. Moreover, this hit on download goodput strongly affects user perceived responsiveness.

Consider the example shown in Figure 1. The goodput over time of an asymmetric link with 2 Mbps download and 192kbps upload rate is shown. The download starts at time 10s and its throughput is shown in black. Between seconds 30 and 45 there is a single upload (dashed line). Between seconds 60 and 85 there is a pair of uploads (dashed lines). The buffer can hold 75,000B (which corresponds to 50 full size packets) in the downlink direction and uplink direction, which is a typical setting. The download can achieve nearly 2 Mbps when sent by itself. However, the presence of an upload causes the throughput to drop so that both upload and download achieve about 200kbps. Giving proper sizes to the buffers lessens the interaction significantly, as in figure 2. This example illustrates the key phenomena about simultaneous TCP downloads and uploads that we are investigating. First, uploads can have a dramatic negative effect on download performance. Second, how the buffer is managed (in packets or in bytes) is important. Third, the buffer size is the key factor in mitigating the negative effects. Fourth, bigger buffers are not always the solution.

Many papers invoke the ACK compression phenomenon to explain download performance drop in presence of bidirectional traffic [START_REF] Balakrishnan | The effects of asymmetry on TCP performance[END_REF][START_REF] Balakrishnan | RFC 3449 TCP performance implications of network path asymmetry[END_REF][START_REF] Brouer | Optimization of TCP/IP traffic across shared ADSL[END_REF][START_REF] Kalampoukas | Improving TCP throughput over two-way asymmetric links: analysis and solutions. SIG-METRICS Perform[END_REF][START_REF] Louati | Handling two-way TCP traffic in asymmetric networks[END_REF][START_REF] Zhang | Observations on the dynamics of a congestion control algorithm: the effects of two-way traffic[END_REF]. In a nutshell, ACK compression refers to the clustering of data packets and ACK packets of both connections, which allegedly compromises TCP connections self clocking by the acknowledgements. These papers focus more on proposing solutions to the problem than actually understanding it. For example, we could not find a paper that considers the influence of link buffer sizes on the magnitude of the problem.

In this paper, we study upload download interference and assert that:

1. ACK compression is not the cause of upload download interference, although it could explain why, in some cases, both the uplink and downlink may be not fully utilized [START_REF] Kalampoukas | Two-way TCP traffic over rate controlled channels: effects and analysis[END_REF];

2. Buffers in bytes are beneficial, especially on the uplink side, although not because of the intuitive reason that they naturally give more precedence to ACKs (as ACKs may fit in a buffer which is otherwise full for data segments1 ). Rather, buffers in bytes are useful because they allow much smaller uplink buffers without hurting the upload when a download is present; this is in turn mandatory to reach a satisfactory utilization of the downlink.

We focus first on the case in which the asymmetric link buffer sizes are measured in bytes (Section 3). In this case, interferences between uploads on downloads are milder than usually thought. This case lends itself well to analysis, and the various parameters involved can be tuned to reduce adverse interferences predictably.

However, in many cases, buffers can keep only a limited number of packets regardless of their size and this generally leads to more marked interactions between uploads and downloads. We study this case by simulation in Section 4 and show that it is more challenging to find settings that would consistently give a satisfactory behavior.

To confirm our findings, we ran performance tests on one of the rare links with buffers limited in bytes, a WiMAX link. The results are given in Section 5.

U d Downlink (C p ) Uplink (C u ) D d A u U s D s D d t p B d B u

Interference between upload and download on an asymmetric link

We consider two TCP (half-)connections on an asymmetric link. The download ACK stream shares the upstream buffer with the upload data packets and vice versa. The TCP end points are connected to an asymmetric link (Fig. 3). The buffers at the heads of the asymmetric link are on different network layer entities than the TCP endpoints, so that the TCP connections continuously probe the link bandwidth and packets may be lost. (If the buffers were on the same entity, the behavior would not be interesting since the TCP connections would refrain from transmitting when the buffer is full.) The TCP connections may use the delayed ACK mechanism whereby an ACK is only sent for every other data packet that is received. We will show later how this factors into our analysis. Considering a tagged data packet D d from the download stream, if we leave aside the delayed ACK mechanism, reception of this segment will trigger the emission of an ACK packet A d . This ACK will enter the upstream queue behind an upload segment D u . When they leave the asymmetric link these packets will engender at least one new data packet D d and a new acknowledgement A u that will enter the downstream queue simultaneously. Thus, it is clear that both connections experience similar round trip times. Note that this holds if the end points of the connections are disjoint, as long as the RTT is dominated by the waiting time in the buffers at the head of the bottleneck (asymmetric) link and the propagation time T p .

For a congestion window, cwnd x , where x refers to uplink or downlink and RTT the round trip time, the connection throughputs are approximately T x = cwndx RTT . Thus, for instance if the 2 connections end up with similar average congestion windows cwnd, then they will obtain similar throughputs.

Pendulum of the outstanding data

Two TCP connections exhibit a hieratic behavior when they share a link in opposite directions. The amount of data in the buffers at both ends of the link often shifts quickly from one side to the other. In essence this oscillation reflects the fact that the amount of data in flight from both connections is slow-varying, so if one queue builds up, the other dips. One example appears in figure 5. This is obtained using the setup in Figure 4; the link capacities are symmetrical (2 Mb/s) with different buffer sizes at both ends of the link: 37500 and 15000 bytes.

Two types of oscillations are visible: first quick oscillations between t ≈ 94 s and t ≈ 96 s. The best explanation for these is ACK compression. We note that it does not always emerge: there is nothing similar between 98 s and 100 s for instance. Our point is that ACK compression is a reality but it's not the most important cause of upload download interference. Let us look at the other wide oscillation starting at t = 96 s. At this time, the buffer in R1 overflows. This causes the download to stop sending packets for a while. Then all upload outstanding data goes into the buffer in R2 which immediately overflows. In the remainder of this section, we model this pendulum of outstanding data.

We consider the downlink and uplink capacities C d and C u , with C d = k×C u (and generally k ≥ 1). Moreover, let d d and d u the number of data bytes standing in the downlink and uplink buffers. We neglect the amount of data in flight on the link and we consider that the ACK size is negligible. Then, as noted in section 2,

RTT = 8 d d C d + 8 d u C u . (1)
Also, d d and d u are directly linked to the congestion windows of the upload and download. The data in flight for each TCP connections is the sum of the data segments waiting in the bottleneck buffer and the ACKs waiting in the reverse direction. For example, every time a download ACK enters the uplink buffer, it experiences a delay of 8 du Cu . So the set of ACKs in this buffer stand for duC d Cu data bytes.

d d = cwnd d - d u C d C u (2)
(and similarly for the upload). Plugging 2 into 1 yields simply

RTT = 8 cwnd d C d (3)
and conversely we have

RTT = 8 cwnd u C u (4)
which implies either that: In figure 5, the system is in case 2 most of the time, and goes into case 1 only when the buffer in R1 overflows and subsequently empties because the download stops for a while. Then we go to case 1 and the buffer in R2 immediately overflows.

Let us consider what decides of the case in which the system operates. We consider for example that d d = 0 (case 1). Then, the download does not occupy the downstream link continuously (or just barely), so that its packets are not well interspersed in the uplink buffer. So cwnd d d u

C d Cu .
Multiplying both sides of this inequality by RTT and using Eq. 4, we get

cwnd d cwnd u 8 d u C d RTT C 2 u
and, as RTT = 8 du Cu , we have:

d d = 0 ⇒ cwnd d C d cwnd u C u ( 5
)
This results is not unexpected: queueing happens on the side of the connection that is more intensely filling its buffer. Similarly,

d u = 0 ⇒ cwnd d C d cwnd u C u .
One important consequence of this remark is that, unless the cwnds are kept proportional at all time, one of the links is not fully utilized. Unfortunately, this is not achievable with TCP on an asymmetric link, as the growth of the congestion windows is the same in both directions since it depends only on the RTT. So it cannot be maintained equal to the ratio of the capacities. Then it is impossible to maintain both (half-) links permanently active. Note that we reach this conclusion without invoking ACK compression.

Buffer size for unidirectional traffic

The minimal size for the link buffers is a good starting point so we recall the rule of thumb to set them. This is not directly linked to upload/download interference, but we remark that, in general, buffers should be of different sizes on both ends of an asymmetric link.

With only unidirectional traffic, the link buffers should be large enough to keep the link busy when there is only a single active TCP connection on the link. The usual rule of thumb for buffer size stipulates that it should be

B x = 2 × t p × C x 8 MTU (6)
packets where C x is the bottleneck link capacity [START_REF] Villamizar | High performance TCP in ANSNET[END_REF] and t p , the end-to-end propagation delay. Note that smaller buffer sizes give satisfactory results when a large number of connections are active concurrently [START_REF] Appenzeller | Sizing router buffers[END_REF], but this is obviously not a generally valid hypothesis for access links.

Asymmetric link with Buffers measured in bytes

We seek to obtain a simple expression for the throughput ratio r between upload and download. To this end, we estimate the congestion windows of both connections.

For an upload, first, two factors may limit cwnd u growth:

1. TCP buffer size W at sender or receiver. If W is smaller than the buffer size plus the amount of data in flight further down, the sender is paced by the rate of acknowledgment reception and there is always a full TCP buffer worth of data in flight;

2. Uplink buffer size: the amount of sender data in flight follows a sawtooth curve that culminates when the buffer overflows. cwnd u oscillates between

B * u +Pu 2
and B * u + P u , where B * u is the uplink buffer space left by the download ACKs and P u is the amount of data in flight on the link, P u = 2 C u t p . Then if ρ is the ratio of ACK size to the segment size, (ρ ≈ 1 30 ),

B * u = B u -ρ cwnd d /2 (7)
if ACKs are delayed, and B * u = B u -ρ cwnd d otherwise. Note that we do not take into account that, in principle, some ACKs may be standing in the downlink queue at the moment the uplink buffer overflows. This is a direct consequence of the remark in section 2.2.

Similarly, the throughput of downloads is limited by the downlink buffer size. In this case, we simply neglect the presence of ACKs in the queue, as ACKs are at the same time a lot smaller than data packets and they are less numerous as we still expect the download to have a higher throughput than the upload.

Moreover, we suppose that the downlink buffer (in R1) is larger than the uplink buffer, which is in fact mandatory for reasonable performance. Then the sawtooth oscillations of the upload cwnd u will be much faster than on the download side. Thus, it is fair to consider that as soon as cwnd d grows bigger than what the system can store, that is the downlink buffer size added to the amount of data in flight P d , then the buffer at R1 overflows.

So we have, if B * d = B d :

cwnd x = W, if B * x + P x > W 3 4 (B * x + P x ), otherwise.
So the ratio r between upload and download throughput is then simply:

r ≈ cwnd d RTT d RTT u cwnd u ≈ cwnd d cwnd u ( 8
)
Where RTT x are the RTTs for upload or download. The approximation that RTT u ≈ RTT d will be valid if they are both dominated by the queueing delays, or if the upload and download endpoints are at a similar "distance" (We do not consider interactions between TCP flows of various RTT). In a nutshell, we show that reasonable buffer sizes, in particular on the uplink side, greatly reduces adverse interference between uploads and downloads. Even more strikingly, the throughput ratio r does not depend on k, it is only marginally linked to C u and C d when computing P u and P d !

Simulations

We use the Qualnet network simulator [START_REF]Scalable Networks[END_REF] with the network topology in Figure 4.

The downlink capacity of the bottleneck asymmetric link is 2 Mb/s downlink and 192 kb/s uplink (from R2 to R1). The asymmetric (bottleneck) link propagation delay is t p = 20 ms or t p = 100 ms. On the left side, which can be seen as the "Internet", the upload and download endpoints are placed respectively 5 ms and 1 ms "away" from the R1-R2 link, so that packets of the two connections do not follow a perfect cycle from one round trip to the next. Each simulation run represents 10 minutes of traffic. The TCP variant is New Reno with selective acknowledgments. Simulation download to upload ratio Ratio estimate q 6.1: 20 ms propagation delay q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 5 10 15 20 . the download to upload throughput ratio obtained by simulation. The vertical line is the downlink to uplink capacity ratio k. Overlying points are moved to form small clusters; circles are points for which the downlink buffer is smaller than the bandwidth delay product. Point sizes are proportional to the TCP buffer size. Red points correspond to link buffer sizes that follow the rule of thumb for a RTT of 300 ms. 7.1: 20 ms propagation delay q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 5 10 15 20 In Fig. 6, to validate our estimate of r, we plot cwnd d cwndu against the obtained download to upload throughput ratios for TCP buffer size in (16384, 32768, 65536, 131072); uplink buffer size in (6000, 9000, 15000, 30000) and downlink buffer size in (22500, 37500, 75000), all in bytes. The dots clearly follow our estimate, so that it is effectively the link buffer sizes that govern the upload to download throughput ratio, provided that TCP buffers are large enough (larger points on the figure) and the downlink buffer is large enough (the circles are offset to the left).

The desired ratio of approximately 10 between upload and download is obtained for TCP buffers of at least 64 kB and link buffers of 9000 B (uplink side) and 75000 B (downlink side). Note that this corresponds to the rule-of-thumb buffer size for approximately 300ms RTT, which is large enough in most cases [START_REF] Huffaker | Distance metrics in the internet[END_REF]. To illustrate the system behavior, we take a closer look at the case when both the uplink and downlink are maintained busy. We focus on the point marked by a dot in Figure 6. Figure 8 plots the congestion windows and buffer occupancies at the start of the simulation. Faster oscillations for the download are evident, as well as the dip in the downlink buffer occupancy after it overflows. To validate that when the uplink buffer overflows, the downlink buffer is mostly empty, we tion: when the upload transmission rate approaches the capacity, then download ACKs will queue in the uplink buffer. They will cause packet drops if the uplink buffer is not large enough to hold them all. For a smaller uplink buffer size, the download will not be able to occupy the entire available downlink bandwidth, as the ratio of the cwnd x will be close to 1.

Using large buffers on both ends of ADSL lines is a common practice [START_REF] Brouer | Optimization of TCP/IP traffic across shared ADSL[END_REF] and this causes a notable drop of the download throughput in the presence of uploads. In other words, most ADSL link operate to the right of Fig. 10; to the point where concurrent uploads and downloads obtain similar throughputs.

Experimentation with a WiMAX link

To confirm experimentally the observation of Section 3, we used one of the rare network links that use buffers limited in bytes: an 802.16d fixed WiMAX system (FDD base station, half duplex subscriber units). We checked that the buffer is measured in bytes by verifying that queueing delay in a full buffer does not change when loading the link with short or long packets.

On this system, we set QoS profiles to limit the maximum information rate (MIR) to 2 Mb/s downlink and 192 kb/s uplink. With these settings, a single TCP download goodput is 1.9 Mb/s, and 0.16 Mb/s for a single upload (goodput ratio of 12). If one download and one upload are present concurrently, then the download reduces to 1.04 Mb/s, while the upload gets 0.15 Mb/s. We note the download rate drops by about a half when an upload is present. However, the downlink is still able to maintain a throughput ratio of 7. Our simulation results suggest that with buffers measured in packets, the goodput ratio would drop to close to 1. This is on of the rare cases where a much higher ratio is maintained by the use of buffers limited in bytes, although this fact is most probably not the cause of this unusual design choice.

Conclusion

In this paper, we showed that TCP performance on asymmetric links depends strongly on the buffer sizes on both sides of the link, as well as on how the buffer size is counted-in number of bytes or packets. Also, we consider that interferences between upload and download are not due to ACK compression.

We conclude that using a buffer limit calculated in bytes allows to greatly reduce adverse interference between uploads and downloads. This is not usual practice, though. On ADSL links, using buffers of adequate sizes counted in ATM cells would most probably be quite straightforward and sufficient. We also recommend to use buffers of relatively small size, especially on the uplink side, which is again not usual.

In the case of buffers that have a capacity set to a given number of packets, the best solution is to use fair queueing mechanisms, or set the buffer to a reasonable size to avoid either uploads starvation or drastic download goodput lessening. Also, simply restricting TCP buffer size for the uploads can limit uplink buffer occupancy, although this is a limited solution when several uploads coexist.

Figure 1 :Figure 2 :

 12 Figure 1: Interference between one download (solid line) and uploads (dashed lines: one then two) with buffers of size 75,000 B on both ends of the asymmetric link

Figure 3 :

 3 Figure 3: Two flows in opposite directions on the same link. The data packets (thick lines) in a buffer interact with the opposite direction's ACK packets (thin lines). Both flows experience the same delays over a round trip.

Figure 4 :

 4 Figure 4: Simulation setup

1 .

 1 d d = 0 and then RTT = 8 cwndu Cu ; or, 2. d u = 0 and then RTT = 8 cwnd d C d ; or, 3. d u = 0 and d d = 0 only if cwndu Cu = cwnd d C d .

Figure 6 :

 6 Figure 6: Ratio estimate cwnd dcwndu vs. the download to upload throughput ratio obtained by simulation. The vertical line is the downlink to uplink capacity ratio k. Overlying points are moved to form small clusters; circles are points for which the downlink buffer is smaller than the bandwidth delay product. Point sizes are proportional to the TCP buffer size. Red points correspond to link buffer sizes that follow the rule of thumb for a RTT of 300 ms.

Figure 7 :

 7 Figure 7: Ratio estimate cwnd d cwndu vs. the download to upload throughput ratio obtained by simulation, 10 concurrent upload connections.

Figure 8 :

 8 Figure 8: Outstanding data (cwnd) of upload, download and buffer occupancy at R1 and R2.

Figure 10 :

 10 Figure 10: Buffer measured in packets, TCP buffer size 64kB. Each point is 2000s of simulation; the horizontal line is k.

  We will see that this is what happens in most cases.

	Outstanding data
	92	94	96	98	100	102
	Queue size
	92	94	96	98	100	102
			Time (s)
			5.1: Buffer occupancy at R1
	Outstanding data
	92	94	96	98	100	102
	Queue size
	92	94	96	98	100	102
			Time (s)
			5.2: Buffer occupancy at R2
	Figure 5: Buffer occupancies at both ends of a symmetric link with different
	buffer size (37500 and 15000 bytes)

h t t p : / / r r . l i g l a b . f r I S S N 2 1 0 5 -0 4 2

As discussed in http://www-nrg.ee.lbl.gov/floyd/REDaveraging.txt, cited in RFC

2.1 ACK compression is a side effect of upload/download interactionSubsequently, because TCP congestion windows are inflated upon reception of an acknowledgement, data packets are produced back-to-back. When this process repeats, then ACK compression appears. Note that in this perspective, it does not matter if ACKs are compressed or not for uploads to impact downloads. Consequently, simply addressing this aspect by pacing TCP senders does not help. In fact, RFC 3449 explicitly rules out this type of solution.On the other hand, prioritizing ACKs in the upstream buffers reduces the RTT of downloads, thus effectively allowing downloads to occupy the whole downlink bandwidth. Using any fair queueing mechanism at the uplink helps to solve this problem, as ACKs represent little volume compared to the upload data segments, so that they are effectively prioritized.

plot the convolution of the buffer occupancy time series in Fig. 9. The negative correlation is patent. We also would like to point out that the analysis in Section 3 can be extended to the case of multiple uploads, which is relevant in the case of P2P traffic for instance. Fig. 7 corresponds to this case, with 10 upload connections instead of one. The results are very similar to what we obtain with a single upload.

Uplink buffer measured in packets

In most operating systems (FreeBSD, Linux. . . ) and for e.g. Cisco routers as far as the authors can tell, buffers can only hold a limited number of packets regardless of their size. So having buffers measured in packets is the usual case. Unfortunately this is not as simple to model as for buffer size in bytes.

In particular, let's look at the minimum size for the uplink buffer. When there is one upload and one download, for each data packet sent uplink, n ACKs corresponding to k download data segments may concomitantly enter the uplink buffer. Consequently, when the upload TCP sender inflates its congestion window, its data packet compete with numerous ACKs. Thus using a buffer of e.g. 6 packets (9000 B) as in the previous section leads to a high loss rate that would cause the upload to stall.

In Fig. 10, we display the throughput ratio between upload and download for various uplink buffer size. Clearly there is a range of buffer sizes that would give satisfactory results, but it depends on the number of ACKs sent per TCP segment. With delayed ACK, in Fig. 10.1, one acknowledgment is sent upon receipt of every other segment; whereas there is one ACK per segment otherwise (Fig. 10.2). Without delayed ACK, the upload to download ratio shows strong variations for adjacent uplink buffer sizes. This variability is not due to the simulation duration, but rather to the fact that in this region uploads are starved, and their behavior is hard to predict or capture in this kind of adverse conditions.

Empirically, the uplink buffer should be at least half the size of the downlink buffer, or uploads will starve. This has a relatively straightforward explana-


M Baumgartner

F Kameier

J Hourmouziadis

Order Non-Engine

Blade

Non-Engine Order Blade Vibration in a High Pressure Compressor

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

A major task in the development programme of a multistage compressor is to ensure that the blades have no severe vibration problem. This definition depends upon the stress criteria, which are linked to the fatigue properties of the component. Although in the design phase much effort is being undertaken to produce blades with a satisfactory mechanical integrity it is limited by the fact that the excitation forces are difficult to predict. Usually the design work concentrates on the calculation of frequencies and the stress distributions associated with particular modeshapes. The standard procedure is to generate a Campbell-diagram, where the frequencies for different running conditions are plotted versus rotor speed. The intercepts of engine order spokes with the frequency lines give the information required to make a decision whether the particular design is acceptable or not. The final assessment of the complete dynamic picture, however, is taken from engine test runs. With the knowledge of the predicted frequencies the vibration modes are able to be identified from the measured frequencies. The most important job is to determine whether the magnitude of the highest vibration levels is below the admissible limits set. Normally the highest amplitudes occur when an engine harmonic coincides with an eigenfrequency of the blade. In this case the blades are being excited by unsteady aerodynamic forces caused by a non-uniform pressure field due to wakes of upstream vanes, potential effects of downstream vanes or further unsymmetrical flow conditions. Sometimes, however, high vibration amplitudes cannot be attributed to engine orders. This requires a careful examination because the vibration might be associated with stall or acoustic resonance in the duct.

In this paper an experimental investigation about the aerodynamic sources which lead to high non-engine order vibration on the first stage of a multi-stage high pressure compressor are presented. The unsteady pressure and velocity fluctuations are determined by pressure transducers mounted at the casing wall just upstream of the first rotor and by a hot-film probe at the trailing edge region of the blade adjustable in the radial direction. The blade response is measured by strain gauges fitted on the aerofoil.

Classification and Evaluation of Flow-Induced Vibrations

Amongst other tasks the prediction of frequencies and mode-shapes is a standard procedure when designing compressor blades. A Campbell-diagram in combination with design criteria is a powerful tool to get a clear picture of the dynamic behaviour of the blade. The intercepts of frequency and engine order lines tell at what engine speed the forced vibration is about to occur. The aerodynamic excitation and the mechanical damping properties that affect the resonanceamplitudes are not covered.

Self exciting mechanism like flutter, acoustic pressure waves or rotating instabilities, however, do not even allow to predict the onset of high amplitudes. Often they occur in an experimental engine-test programme when the stability boundaries of the compressor are being determined.

Flow Induced Vibration

Forced Vibration

The blades rotate through a steady flow field that is non-uniform in circumferential direction. This flow field is being experienced by the blades as cyclic pressure fluctuations varying in time. Strain gauges fixed to cantilevered blades will observe the response of the blades as a series of sinusoidal excitations according to a Fourier-decomposition of the excitation signal. Resonance occurs in the case of an eigenfrequency of the blade coinciding with any of these harmonics, which is an integer multiple of the rotational speed.

The dynamic behaviour of the coupled blades is different compared to cantilevered blades [START_REF] Thomas | Zur Frage der Gefährlichkeit von Paket-oder Einzelschwingungen bandagierter Turbinenschaufeln[END_REF]). The whole bladed disk assembly forms complex vibration modes. Every classical mode like 1st flap (1F), 1st torsion (1T) is, as it were, an upper term for a complete family of individual modes, which are classified as nodal diameters (ND). The maximum number of nodal diameters is usually half of the number of blades (N) per stage. Whereas the blades perform an in-phase vibration mode for a zero ND, they vibrate out-of-phase for a N/2 ND [START_REF] Ewins | Structural Dynamic Characteristics of Bladed Assemblies[END_REF].

Since for tuned cyclic systems every mode appears in pairs with orthogonal mode-shapes resonance conditions are only possible if the excitation forces are orthogonal as well. As a result of this a backward travelling wave builds up. This is the most significant difference to a cantilevered blade resonance condition:

A forced vibration resonance is only possible if the integer multiple of the rotor speed (Engine Order EO)

has the same integer value as the nodal diameter (ND).

The pressure distribution in the circumferential direction can be regarded as a series of sinusoidal waves for any frequency. So the excitation frequency of say a 5th EO also contains, besides the wave with 5 lobes, also waves with 3,4,6 etc. lobes. Thus an aerodynamic wave with 6 lobes having the frequency, which is exactly the 5th EO, may excite the 6thND structural bladed disc mode if the frequencies fit.

This may not count for the standard forced response case where the 5th aerodynamic ND is the strongest for a 5th EO. However the aerodynamic ND has to coincide with the structural ND and not the EO. Therefore at a non-engine-order resonance condition the aerodynamic and structural ND are the same.

For rotors with coupled blades it is necessary to distinguish between strict periodic excitation and periodic excitation. Strict periodic excitation only yields to integer multiples of the first harmonic wave lobe-number. Periodic excitation with small disturbances of the pressure field results to various waves having all possible aerodynamic ND.

Self-Excited Vibration

In contrast to forced vibration the alternating force disappears when the vibration stops. The alternating force is controlled by the motion.

Acoustic Resonance

Acoustic pressure fluctuations with a propagation speed being the speed of sound can cause high level vibrations of mechanical structures. There is only one frequency of an acoustical resonance and its wave length has to fit with a geometrical length to generate high pressure levels. Bleed cavities in turbomachines have this kind of resonance.

The frequency depends only on the geometry and not on the flow velocity. Similar to the sound radiation of a Helmholtz-Resonator the frequency is influenced by the resonator volume, the speed of sound and the diameter of the neck or the hole in the resonator body [START_REF] Fletcher | The Physics of Musical Instruments[END_REF]. The excitation is a function of the vortex shedding frequency, which is caused by the tangential flow across the inlet hole. [START_REF] Parker | An Experimental Investigation of the Generation and Consequences of Acoustic Waves in an Axial Flow Compressor: Large Axial Spacing between Blade Rows[END_REF] also conclude that the source of an acoustic resonance excitation in turbomachines is due to vortex shedding from the rotor or stator blades or even from support struts.

Vortex shedding, however, does not necessarily need to excite an acoustic field. Particularly in the immediate vicinity of the blade pressure fluctuations can be generated the propagation speed of which differs from the speed of sound.

Rotating Flow Instabilities

The best known rotating flow instability in turbomachines is rotating stall. The increasing of the incidence angle of a stage leads to a boundary separation which due to flow non-uniformities sets on at one or more circumferential positions. A stalled blade results in a flow diversion that tends to overload one adjacent blade and to unload the other adjacent blade. This causes the overloaded blade to stall and to unload the originally stalled blade, thus generating the stall cell to move around the blade row with a smaller velocity, which is about half of the rotor speed. [START_REF] Greitzer | Review -Axial Compressor Stall Phenomena[END_REF]. Pressure transducer can measure that pattern leading to a discrete frequency with harmonics of it.

Rotating stall can be regarded as a special case of a rotating aerodynamic instability reported by [START_REF] Kameier | Experimentelle Untersuchung zur Entstehung und Minderung des Blattspitzen-Wirbellärms axialer Strömungsmaschinen[END_REF] and Kameier and Neise (1995/II). The combination of flow recirculation due to an enlarged tip clearance and a boundary separation causes a vortex shedding in the tip region of the rotor blades which leads to high pressure fluctuations and might cause high level blade vibration. Unlike the rotating stall the pressure fluctuations are unsteady in a frame rotating with this instability pattern. Fig. 2 illustrates this effect. An observer moving with the instability cell would measure a discrete frequency signal associated with the pressure fluctuation whereas moving with the rotating stall cell he would not detect any signal due to the stall. In a non-rotating coordinate system the rotating instability would be measured as a range of frequencies with a characteristic frequency signature different to the rotating stall which would be received as a single discrete frequency with possible harmonics of it at the most.

One can imagine rotating instability as a rotating loudspeaker that is moving relative to the rotor blade in the circumferential direction.

Flutter

A number of investigations and explanations about flutter have been published about the different kind of flutter and excitation mechanism. Flutter is a flow induced instability at or close to the eigenfrequency of the rotor blade. The frequencies of flutter are not an integer multiple of the engine speed. Because even small stochastic vibration levels are sufficient to build up high amplitudes no periodic external excitation source is required. The input-energy from the fluid motion to the blade is higher than the dissipationenergy provided by the mechanical damping.

Whereas it is possible to accelerate or decelerate through the speed range of an engine order vibration, the flutter amplitudes can only be reduced by backing off in speed.

A literature survey shows that plenty of papers are being published every year. A detailed overview about various aspects in flutter and a classification is given by [START_REF] Fung | An Introduction to the Theory of Aeroelasticity[END_REF] and [START_REF] Sisto | Introduction and Overview, AGARD Manual on Aeroelasticity in Axial-Flow Turbomachines[END_REF].

Theoretical Formulation of the Rotating Source Mechanism

In a fixed frame of reference a one dimensional description of the unsteady azimuthal pressure fluctuation in an annulus p( F ,t) can be thought as a series of sinusoidal waves with different lobes  across the circumference having time-dependent amplitudes A and phase angles . Since the timedependent values can be expressed as continuous spectra, the pressure fluctuation follows as [START_REF] Tyler | Axial Flow Compressor Noise Studies[END_REF])

    p t A t F n F F n F n F n        , c o s           1 (1)
where the amplitudes are schematically illustrated in Fig. 3. with that frequency. Assuming the source is rotating relative to the rotor with an angular velocity of  S F then in a frame of reference rotating with the source (S) the pressure distribution can be given in the same way as in equation [START_REF] Thomas | Zur Frage der Gefährlichkeit von Paket-oder Einzelschwingungen bandagierter Turbinenschaufeln[END_REF].

    p t A t S n S S n S n S n        , c o s           1 (2)
Similar in a coordinate system fixed with the rotor

    p t A t R n R R n R n R n        , c o s           1 (3)
The transformation of the frequencies from the source frame to the fixed frame (F) and to the rotating frame of reference (R) follows with the geometric relations given in Fig. 4.

  F S S F t    , ( 4
)   F   R R F t  , ( 5
)   S R R S t    , (6) R F S F R S      . ( 7
)
In equation ( 4) the angular coordinate in the fixed frame of reference  F is the sum of the angular coordinate in the source frame of reference  S and the angular distance of the source frame of reference relative to the fixed frame of reference  S F .

Because the rotating source will be seen from the rotor system, the equations ( 6) and ( 7) can be modified with the relation

R S S R     (8) to   S R S R t    and (9)    R F S F S R   . ( 10
)
Substituting equation ( 4) into ( 2)

      p t A t F n F F n S S F n F n          , c o s            1
(11) allows to determine the frequencies of the rotating source frame in the fixed frame by comparing ( 11) with [START_REF] Thomas | Zur Frage der Gefährlichkeit von Paket-oder Einzelschwingungen bandagierter Turbinenschaufeln[END_REF]. For the aerodynamic modes vibrating with the source frequency  SM S the following relation can be given

   SM F SM S S F    . ( 1 2 )
In the same way the substitution of equation ( 9) into ( 2) results in a formulation of the rotating source frequency  SM R in the rotating system

   SM R SM S S R    . ( 1 3 )
Note that the -values might be positive and negative according to the equations ( 1),( 2) and (3).

Equations ( 12), ( 13) and (10) result in an expression for the aerodynamic nodal diameters

     SM F SM R R F  . ( 14
)
(14) and ( 12) allow the frequency of the pressure wave in source system to be calculated:  

 S F t  S  F  R S o u r c e f r a m e R o t o r f r a m e  R F t  R S t
    SM S SM F SM F SM R S F R F      ( 15
)
For two adjacent  the equations ( 12) and ( 13) lead to the angular velocity of the rotating source

 S F SM F SM F        ( ) ( ) 1 . ( 16
)
The rotating stall flow pattern does not generate frequency components in the frame of reference which is rotating with the stall. Therefore  SM S  0 and from the equations ( 12) and ( 13) follow for the fixed and rotating frame of reference

  RS F RS F   (17)
and

  RS R RS R   . ( 18
)
A comparison between equations ( 17) and ( 18) with equations ( 12) and ( 13) shows that the rotating stall is a special case of the rotating instability.  RS The equations ( 11), ( 12) and ( 13) show for a positive source frequency  SM S  0 the relationship

 SM F  0 for   0 or      SM S S F /    SM R  0 for   0 or     SM S S R /  
According to equation ( 1)  is always positive and negative, therefore it is necessary to consider that the frequencies can be negative. Because the measurement devices always give positive frequencies equation ( 14) has to be rewritten to , whereas the rotating stall would only cause a constant pressure distribution thus having a particular pressure wave with a certain  . The transformation into the fixed frame leads, according to equation ( 12), to a straight line containing different frequencies for the rotating instability but only to a single frequency for the rotating stall. Because of equation ( 16) the difference between adjacent peaks is constant, which is a characteristic of rotating instability when interpreting measured pressure data in the fixed-frame, Fig. 5 also shows that any aerodynamic nodal diameter can excite a cantilevered blade mode, whereas a resonance for a bladed disc is only possible if the mechanical and aerodynamical nodal diameters are the same. For  SM S  0 and  S F  0 equation ( 10) and ( 13) lead to forced response conditions.

     SM F SM R R F  . (19)
    -1 0 1 2 3 4 5 6 7 8  SM S  SM F Fixed Frame Rotating Instability  S F  S F  RS F Fixed Frame Rotating Stall Source Frequency   SM R Rotating Frame Rotating Instability  S R  S R   SM R Eigen Bladed Disk  ( )   SM R Eigen Cantilever  ( )   ND

Blade Design, Test-Rig, Experimental Set-up and Measurement Technique

Blade Design

Fig. 6 shows the 1T mode-shape of the first rotor blade of a multi-stage high pressure axial compressor. The aerofoil is coupled by means of a snubber with the adjacent blades. This is necessary to avoid a low engine order excitation within the running range because the blade features a high aspect ratio and thin profile thickness design. Strictly speaking the plotted mode-shape is the analytical solution for the 8thND.

The mode-shape for the entire assembly is displayed in Fig. 7.

Test-Rig

Experimental Set-up

The pressure fluctuations on the casing wall have been monitored by using Kulite XTE 190 pressure transducers. They are mounted flush with the inner wall, upstream of the blade leading edge of the first stage ( Fig. 9).

A single hot-film probe (Dantec 55R03 Fiber-Film-Sensor) has been employed to measure the velocity fluctuations in the blade tip trailing edge region of the first rotor. To vary the radial measurement position the probe is able to be moved radially from the casing.

The stress measurements have been carried out with standard strain gauges measurement technique. The signal is being transmitted through a slip-ring from the rotational system to the fixed system.

Measurement Technique

The complete transient data information has been recorded on a multi-channel tape recorder. A HP 9000 UNIX workstation has controlled the data acquisition and the processing unit HP 3565 with anti-alias protection, FFT and averaging processor on each channel. A standard multi-channel FFT-Analyser software has been applied for evaluating the recorded data. Several replays from tape allow nearly infinite resolutions in frequency.

To display the frequency signature of a component against engine speed or against time the Z-Mod plots are used. They are similar to waterfall plots except that each spectrum is displayed as a single vertical line with amplitudes being represented by a colour scale (in this paper for copy reasons by a grey scale). For minimising noise effects each plotted spectrum in the Z-mode is an average of 10 measured spectra. The frequency spectra contain 1600 lines with a span of 12.8 kHz that is a frequency resolution of 8 Hz.

Test Analysis and Interpretation 5.1 Comparison of Pressure Fluctuations and Blade Vibration

Fig. 10 shows for a slow acceleration the Z-Mod plots versus speed of the pressure transducer in the casing wall (top) and the blade strain gauge (bottom). The upper diagram displays a frequency span of 8.5 kHz for the wall pressure and the lower diagram a frequency span of 3 kHz for the blade vibrations. The blade passing frequency is clearly visible on the Kulite picture as a thick 31st EO line. Between 89.5 -96.5 % rotor speed high wall pressure fluctuations occur. Also at this speed range high blade vibration amplitudes for the 1F and 1T modes build up the maxima of which do not intersect with the engine order lines. More detailed information of this effect is given by individual spectre plots at engine speed of 85%, 94% and 101% (Figure 11, 12, 13). These pictures illustrate the correspondence of blade vibration and wall pressure fluctuations.

At 85 % speed (Figure 11) the wall pressure spectrum shows harmonics of rotating stall frequencies (equation 22) and the first blade passing frequency (1st BPF). The lowest rotating stall frequency is larger than the rotor shaft speed (1st EO). This presumably results from an asymmetric distribution of two stall cells that cause the 3rd and 4th frequency peaks. The blade passing frequency is being picked up by the pressure transducer because the passing blades generate an oscillating potential pressure field. The levels are obviously more dominant than the pressure field of the rotating stall wave. Thus the stall is more likely to be a part span stall rather than a full span stall, which would have resulted in higher levels due to flow separation along the entire blade. The blade vibration shows low activity for the 1st bending and 1st torsion mode frequencies.

At 94.3 % mechanical speed (Figure 12) high blade vibration amplitudes for the 1F and 1T modes build up. The pressure spectrum shows large fluctuations. The peaks, however, are still smaller than the blade passing frequency. The frequencies are in the range of about half the blade passing frequency. The blade vibration frequencies neither correspond to integer multiples of the engine speed nor fit to the frequency components of the pressure fluctuations.

For higher compressor speed (Figure 13) the blade vibration amplitude levels fade away. Simultaneously the levels of the pressure fluctuations disappear. Only multiples of the rotor speed with the highest amplitudes for the blade passing frequency are visible.

Explanation of Pressure Fluctuation Frequencies

In addition to Fig. 12, Fig. 14 details the frequency pattern of the pressure field with an increased frequency resolution at about 94% mechanical speed. This picture shows constant distances between the peak amplitudes. This would confirm equation ( 16).

The distance of the spikes is 155 Hz that is 62% the rotor speed. The measured frequencies behave according to equation ( 12 measured frequency pattern: A source frequency in a rotating frame, which is able to excite pressure waves with various nodal diameters, leads to a similar frequency behaviour in the fixed frame (Kameier and Neise, 1995 II).

The data analysed in this investigation show the same phenomenon of a rotating instability: A vortex shedding mechanism propagating relative to the rotor excites pressure waves with a certain frequency that can excitate rotor blades to high amplitudes.

Nodal Diameter and Source Frequency of Rotating Instability

To calculate the azimuthal mode number  according to equation ( 19) both the frequencies of the fixed and rotating frame are required. The frequencies of the fixed frame  SM F can be taken from Fig. 14 whereas due to the lack of unsteady pressure instrumentation in the rotating frame only frequencies measured by strain gauges  SM R can be employed. The 3rd and 4th columns of Table 1 give results for the positive and negative sign in equation ( 19). The source frequency  SM S is being calculated by equation ( 12) considering that  and  SM F might also be negative. Non-integer numbers of  are caused by inaccuracies of the frequency-data analysis, the lack of unsteady pressure instrumentation or non possible frequency combinations.

From a bladed disk analysis the mechanical nodal diameter was identified to be a 5th or 6th ND for the 1T family and a 7th or 8th ND for the 1F family. This would confirm the determination of the aerodynamic ND of the 3rd column in Table 1. The data of column 4 do not fit to the bladed disk eigenfrequencies.

 R F Hz 2 249 π  .   SM F 2 [Hz]   SM R 2 [Hz]      SM F SM R R F       SM F SM R R F    SM S / 2 [Hz]  S F 2 [Hz]  (-)  (+) 3503 1911 6.4 21.7 +2512 +4494 +133 +6873 +155 3348 1911 5.8 21.1 +2453 +4243 +74 +6622 +155 3193 1911 5.1 20.5 +2395 +3991 +16 +6370 +155 3503 1372 8.6 19.6 +2176 +4830 +468 +6538 +155 3348 1372 7.9 19.0 +2118 +4578 +410 +6286 +155 3193 1372 7.3 18.3 +2059
The results show that a particular aerodynamic ND excites the same structural ND at a non-engine order resonance condition. The frequencies and the nodal diameters of the aerodynamical and mechanical waves are the same.

Measurement of the Radial Distribution of the Rotating Instability

To localise the radial extension of the rotating instability pattern hot-film measurements were carried out. At constant operating conditions the probe was radially moved downstream of the rotor close to the trailing edge. The position has been varied 40 % of the blade span from tip to hub.

Hot-film measurements near blade tips are subject to large measurement errors because close to the rotating blades both the magnitude and the angle of the mean velocity vary periodically by significant amounts. Therefore the results in Fig. 16 are displayed as spectra of the coherence function between the hot-film probe velocity fluctuations and the unsteady pressure probes in the casing wall

         2 21 21 11 22    G f G f G f G f * , (20)
where G 11 ,G 22 are the power spectra and G 21 ,G 21 * conjugate complex cross-power spectra.

Apart from the blade passing frequency (BPF) of the 1st and 2nd stage, frequency peaks due to random fluctuations of the turbulent flow are quite large. In order not to lose the signal of the rotating instability effects these turbulent frequency spots were not averaged out.

Closest to the casing wall (upper spectrum, only half of the hot-film is in the flow) the velocity fluctuations are very noisy. This depends upon the installation of the hot-film probe in a hole where flow separation at the sharp edges occurs. At 91% blade height the rotating instability frequencies are clearly visible with high coherence levels. The levels are dying away with further movement of the hotfilm probe to the centre. Eventually at 65% blade height the RI is no more detectable. The RI only occurs between the blade tip and the snubber. More studies are needed to investigate the impact of the snubber geometry on the RI phenomenon.

A further comparison of the total pressure distribution along the blade height indicates significant flow variations in the blade tip region at high pressure ratios especially at off-design conditions (94 % rotor speed). A continuous break down of the total pressure near the blade tip for an increasing pressure ratio is plotted in Fig. 17 The unsteady measurements as well as the steady measurements of the total pressure distribution along the blade height confirm the supposition of the

Figure 1 :

 1 Figure 1: Classification of flow-induced vibration

Figure 2 :

 2 Figure 2: Schematic presentation of rotating stall and rotating instability pressure variation at different times

Figure 4 :Figure 3 :

 43 Figure 4: Rotational direction of the source frame and the rotor frame relative to the fixed frame of reference.

  velocities of the rotating stall cell in the fixed and rotating frame of reference. Since  RS F is usually smaller than  R F but propagating in the same direction, the sign of  RS R

Fig. 5

 5 Fig. 5 explains how the frequencies and aerodynamic nodal diameters relate in the source frame and in the fixed frame. In the source-frame the source-frequency  SM S can, according to equation (2), excite numerous pressure waves       

Figure 5 :

 5 Figure 5: Schematic of frequencies and aerodynamic nodal diameters

Fig. 8

 8 Fig. 8 gives an outline of the 10 stage high pressure compressor test rig facility that is being driven by a 18 MW electric motor. The test compressor has a

Figure 6 :

 6 Figure 6: 1T Modeshape (1 T) of the first rotor blade Figure 7: Mode shape for the entire assembly of a multi-stage high pressure axial compressor

Figure 8 :

 8 Figure 8: Schematic of the 10 stage high pressure compressure.

Figure 9 :

 9 Figure 9: Experimental set-up of the Kulite XTE 190 transducer in the casing wall.

  ) or the straight line in the fixed frame of Figure5. The results tie up very well with results by Kameier and Neise (1995 I) on a low speed fan, where similar frequency patterns have occurred. (Fig.15). A more thorough investigation with a rotating loudspeaker confirmed the characteristic of the

Figure 10 :

 10 Figure 10: Comparison of blade vibrations and wall pressure fluctuations.

Figure 11 :

 11 Figure 11: Comparison of wall pressure and blade vibration spectra, N=85 %.

Figure 12 :

 12 Figure 12: Comparison of wall pressure and blade vibration spectra, N=94 %.

Figure 13 :

 13 Figure 13: Comparison of wall pressure and blade vibration spectra, N=101 %.

Figure 14 :

 14 Figure 14: Power spectrum with an increased frequency resolution, N=94 %.

Figure 15 :

 15 Figure 15: Pressure spectra at the casing wall, upstream the leading edge of the rotor; upper diagram: low-speed compressor (Kameier 1994); lower diagram: first stage of a multi-stage high-speed compressor of the present investigation.

  .

Figure 16 :

 16 Figure 16: Coherence of pressure fluctuations at the casing wall (upstream leading edge of rotor 2) and velocity fluctuations (trailing edge of stator 1) for various blade heights, N=94%

Table 1 :

 1 Calculated aerodynamic nodal diameters  and source frequencies  SM S  0 , rotor speed

Improvements of Aerodynamic Blade Loading

To reduce high aerodynamic blade loads in the tip region the first stage was modified. The aim was to move the operating conditions to greater incidence angles to achieve for the required mass flow rate the highest possible reserve against the operating conditions sensitive to instabilities. Fig. 18 shows for a cross-section in the blade tip area the new design allows more incidence and higher pressure ratio. Not only at design conditions but also at off-design conditions the flow stall begins at higher angles of attack.

Conclusions

An experimental investigation on a high pressure multi-stage compressor shows that for a certain operating condition high vibration levels on the first stage blades exist. The amplitudes are not in resonance with engine orders. Pressure fluctuations in the casing wall exhibit a characteristic pattern.

A theoretical model describes the effect: A sound source propagates relative to the rotor. It leads in its own frame of reference to an excitation of a series of sinusoidal pressure waves. The waves can be measured in the fixed and in the rotating system. The source has a particular frequency and is presumably caused by vortex shedding at off-design conditions. Resonance occurs if the bladed disk modes and the pressure waves have the same frequencies and the same nodal diameters. Because the source is moving relatively to the rotor the resonance happens at nonengine order conditions.

The frequency characteristic agrees well with results obtained on a low pressure fan experiment, where the propagation of the rotating source was investigated more in detail.

The rotating instability is similar to the rotating stall but the latter does not excite pressure fluctuations in its own frame of reference thus does not lead to the characteristic frequency range measured with pressure transducers.

Hot-film measurements have shown the radial dependency of this rotating instability phenomenon. In the investigated case it only occurs between blade tip and the snubber similar to a part span stall. Measurements of total pressure distribution along the blade span have also confirmed the effect.

Nomenclature


Rachel Bawden

Guillaume Wisniewski

Hélène Maynard

Investigating gender adaptation for speech translation

Keywords: Traduction de la parole, TA, genre, adaptation, corpus parallèle Speech translation, SMT, gender, adaptation, parallel corpus

In this paper we investigate the impact of the integration of context into dialogue translation. We present a new contextual parallel corpus of television subtitles and show how taking into account speaker gender can improve machine translation quality in terms of BLEU and METEOR scores. We perform a manual analysis, which suggests that these improvements are not necessary related to the morphological consequences of speaker gender, but to more general linguistic divergences.

RÉSUMÉ

Étude de l'adaptation au genre du locuteur pour la traduction de la parole Dans cet article nous évaluons l'impact de la prise en compte du contexte dans la traduction de dialogues. Nous introduisons pour cela un nouveau corpus parallèle, issu des sous-titres de séries télévisées, comportant de nombreuses informations contextuelles. Nous montrons comment la prise en compte du genre du locuteur permet d'améliorer la qualité de la traduction automatique en termes de score BLEU et METEOR. Une analyse manuelle montre toutefois que ces gains ne sont pas nécessairement liés aux conséquences morphologiques du genre du locuteur, mais à des différences linguistiques plus générales.

Introduction

Journalistic and planned texts have long been the focus of attention in the Statistical Machine Translation (SMT) community largely due to the availability of large parallel corpora from parliamentary debates and the relative lack of data in other domains. Over the last decade, more and more emphasis has been placed on speech translation (Hardmeier, 2012), made possible thanks to the availability of speech-like parallel corpora such as the Ted talks and parallel subtitles (Lison & Tiedemann, 2016). 1,2 The translation of speech-like texts is a challenge for standard SMT systems (trained on journalistic or parliamentary corpora) due to the very different nature of the texts; linguistically, spoken sentences are often ungrammatical, incomplete and contain a greater degree of lexical diversity, but they must also be situated within a dialogue context, in which prosody, common ground and speaker information can offer the only ways of providing a correct translation. One such example is that of French gender agreement, in utterances such as "I am tired", translated as "Je suis fatiguée" for a female speaker and "Je suis fatigué" for a male speaker, for which speaker gender is the only possible way of determining the correct adjectival suffix. 3 In this work, we aim to evaluate the impact of integrating external contextual information on translation quality. We present two contributions: (i) the creation of a contextualised parallel corpus of spontaneous dialogues, taken from television subtitles adapted from the TVD dataset (Roy et al., 2014) and (ii) exploratory experiments on adapting translation systems to one example of contextual information: the gender of the speaker. We provide preliminary results for the translation of English to French subtitles, using automatic evaluation metrics, as well as a manual evaluation of improvements.

Statistical approaches to machine translation heavily depend on training data used, and therefore adapting systems to specific corpora is a very natural move. The choice of data used to tune model parameters is also highly important, as shown by Pecina et al. (2012) when tuning an out-of-domain model using in-domain data. Gender-dependent modelling is a common technique used in speech recognition to adapt models to acoustic differences between men and women's speech (Wahlster, 2000). However model adaptation is not restricted to speech recognition; Kübler et al. (2010) use dialogue act tags to improve a PoS tagger, and in machine translation, adaptive modelling has been used to create topic-dependent (Foster & Kuhn, 2007) and sentence-type-dependent (Finch et al., 2009) models by training and tuning on class-partitioned data.

Our paper is organised as follows. We first describe our TV series corpus production process (Section 2). In Section 3 we present the baseline models, based on pre-existing corpora. In Section 4 we suggest and evaluate a number of adaptations of the baseline models to gender-specific data. Finally, Section 5 provides a more detailed analysis of these changes through a contrastive manual evaluation between gender-specific and baseline models.

The Big Bang Theory reproducible corpus

The TVD corpus: We focus on the translation from English to French of a small but contextually rich subtitle corpus: the first two seasons of the American television series The Big Bang Theory. The dataset is generated using the TVD plugin (Roy et al., 2014), developed to provide reproducible datasets, exploiting visual, auditory and textual data, directly extracted from DVDs and freely available web sources. 4 This is an important addition to primarily textual resources such as OpenSubtitles, which provide little extra contextual information. Amongst the numerous contextual elements available (speech turns, audio signal and images), we choose, in this preliminary work, to focus on and integrate one aspect of this contextual information into translation -the gender of the speaker.

Extracting an English-French parallel subtitle corpus: To use the corpus for the purpose of translation, we first extracted an enriched parallel corpus from The Big Bang Theory TVD plugin. The text is taken from the official OCR-extracted subtitles, 5 since, unlike the manual transcriptions, they exist in multiple languages and are official translations. Following some correction of OCR errors using a small error lexicon and some manual rules, we align the French and English subtitles using their timestamps. Since there is not always a one-to-one mapping between French and English subtitles, and a perfect temporal alignment is not always present, heuristics are used to concatenate subtitles where necessary in order to create parallel subtitle blocks.

Assigning gender to subtitles: Gender is assigned to subtitles by manually mapping speaker names to their corresponding gender. Speaker identities are provided by manual transcripts,6 which are automatically aligned to the audio signal (Bredin et al., 2014). We then transfer speaker identities to the subtitles, based on transcript and subtitle timestamps. We leave the task of using automatically predicted gender to future work; here we only consider reference genders to test our hypothesis. 7 Division into train, development and test sets: We divide this corpus into three datasets: BBTtrain (the first 40 episodes), used to train translation and language models, BBT-dev (the next 6 episodes), used for tuning and BBT-test (the last 6 episodes), used for evaluation. We partition each set into two subsets, one for each gender. Basic corpora statistics can be found in Table 1. Note that there is a strong class imbalance towards male speakers, who produce approximately 3 ⁄4 of all test sentences. Subtitles corresponding to female speakers are also on average shorter than those for male speakers, and the percentage of out-of-vocabulary tokens compared to the two subtitle corpora (OPENSUBTITLES and BBT-train) is much smaller for female than male speakers, perhaps indicating a less heterogeneous use of vocabulary. Note that these characteristics are corpus-specific.

Baseline systems: testing pre-existing corpora

The first step in translating new data is to see how existing data fares for training a standard translation system. We provide baseline models using two pre-existing parallel corpora: EUROPARL (Koehn, 2005) and, more adapted to our domain, (though less commonly used) the film and television subtitle corpus OPENSUBTITLES (Lison & Tiedemann, 2016). 8 As this corpus is very large, we used the Modified Moore-Lewis (MML) Filtering algorithm (Axelrod et al., 2011) to keep only the 8% of OPENSUBTITLE sentences most similar to BBT-train. We hereafter refer to this set as OpenSubs-mml.

A comparison of corpora is given in Table 1. One side-effect of filtering is the drop in average length between the corpus before and after filtering, most likely an effect of the fact that longer sentences are more different from each other in the two subtitle corpora. Despite this difference, OpenSubs-mml appears to be the most adapted corpus vocabulary-wise to our BBT data, resulting in the fewest out-of-vocabulary tokens. Note also the relative sentence lengths of source and target sentences. Whereas it is generally the case that French translations are longer than the corresponding English ones (as is the case with EUROPARL), the opposite is seen for the subtitle corpora, most probably linked to a shortening during subtitling due to the use of less literal translations and space constraints.

Translation setup: We use the Moses toolkit (Koehn et al., 2007) for translation. Alignments are computed over all three training sets. All language models are 4-gram models with Kneser-Ney smoothing. Parameters are estimated on BBT-dev using kbmira to optimise the BLEU score. For each corpus, a separate phrase table and language model are produced. Multiple language and translation models are combined using the default Moses model combination approach, and are assigned weights during the tuning step.9 Results: We evaluate the different models using two metrics: BLEU, and METEOR. 10 Results are shown in Table 2. As the models are tuned with BLEU, we judge the best model combination to be the one with the highest scores with the second metric METEOR. This model, OpenSubs-mml+EUROPARL also produce the highest scores for both metrics on the test set. It is unsurprising that both the BBT-train and EUROPARL models generalise poorly; the first lacks coverage because of its small size and the second because it is ill-adapted to speech-like data. However adding EUROPARL to OpenSubs-mml, by far the best-adapted dataset, does improve the scores of the latter. Somewhat surprising is the fact that adding BBT-train does not further improve results, and even degrades them slightly, most likely due to overfitting, as indicated by the BLEU scores on the development set.

Gender-based adaptative modelling

Taking our best baseline system (OpenSubs-mml+EUROPARL), we propose a series of adaptations to take speaker gender into account : (i) changing the tuning data, (ii) adding a gender-specific phrase table, (iii) adding a gender-specific language model, and (iv) adding both a gender-specific phrase table and language model. The additional gender-specific models are estimated using BBT-train sentences uttered by either male or female speakers. We distinguish three types of tuning data: 'all' corresponding to the entire BBT-dev, 'female' to sentences by female speakers and 'male' by male speakers. We test each of the models individually on the female and male BBT-test data.

The results (Table 3) show that exploiting speaker gender is useful for translation, with improvements possible for both male and female speakers as shown on the test set. All adaptations provide some improvement in at least one configuration when compared to the baseline score (first row). The highest score seen for the male test set was the combination of a specific language model, a specific translation model and BBT-dev m for tuning (+LM m +TM m /male), with an improvement of +0.17 BLEU.

The improvement between the baseline and contextualised model was greater for female speakers, with an improvement of +1.09 BLEU for the configuration (+LM f /female). When the gender-adapted models are used, (and the baseline for unknown genders), we see an improvement in scores for BBT-dev (24.61 BLEU, 0.433 METEOR) and BBT-test (25.11 BLEU, 0.444 METEOR).

Discussion and analysis of improvements

We base our discussion on the differences between baseline predictions (∅/all) and those from the model that gave the greatest improvements for each gender: (LM m +TM m /male) and (LM f /female).

Given the language pair (English to French), one type of error that we could hope to have corrected is that of gender agreement, particularly for adjectives and past participles, in sentence such as "I am happy", translated as "je suis contente" for a female speaker but "je suis content" for a male speaker. However given the small size of our dataset, the cases of this phenomenon are few; we manually identify 11 cases in the female test set which, given the lexical choice, could have resulted in a correction of gender, only one of which actually resulted in a correction. 11 So what are the improvements down to? As for many statistical systems, the improvements appear to be diverse and specific to the data used. To better understand the differences found between the baseline predictions and those from the gender-adapted models, we manually compared the quality of these two sets of translations and annotated their differences (See Table 4). 12 The most common differences for both genders were in lexical choices, followed by additions and deletions. A change in lexical choice was more often associated with an improved translation than a degraded one (38% vs. 31% for both genders). However the change most linked to an improvement was addition, and conversely, the change most linked to a degradation was deletion; for male speakers, 73% of sentences whose only difference was an addition were improved (82% for females), and 93% of sentences whose only difference was a deletion were degraded (60% for female).

These observations suggest that the difference in BLEU score might result from differences in sentence length; the BLEU metric heavily penalises translation hypotheses that are shorter than the reference and, as shown in Table 1, female utterances are on average shorter than male utterances. It turns out that the baseline model produces translations that are 99.3% shorter than the reference translations for male speakers and 97% shorter for female speakers, whereas the adapted models produce translations 99.8% shorter for male speakers and 98.8% shorter for female speakers. The heightened improvements linked to addition for female speakers and the decreased effect of deletion compared to male speakers may be explained by the class imbalance in the data. The generic data used to tune the baseline model contains three times as much male data as female data, and therefore the baseline model is biased towards male utterances. Any improvements are therefore lessened with respect to the improvements that can be achieved by adaptation for the more sparse female tuning data. Table 4: Manual analysis of differences between baseline and the best gender-adapted models.

Conclusion

We have shown that it is possible to improve manual and automatic evaluation scores when testing our subtitle corpus on baseline and gender-adapted models. A manual evaluation indicates that these preliminary results do not yet enable us to link these improvements to gender-specific linguistic phenomena such as gender agreement. Improvements appear to be due to other specificities of the datasets, such as average sentence length, and the higher gains for female speakers are almost certainly linked to class imbalance in the data. Further investigations will be needed to fully understand our results. We also intend to extend the work to a larger and more balanced dataset, in order to see whether further improvements can be made. Automatically identifying speaker gender will also enable us to forgo the need for manual transcriptions. We will also turn to other types of contextual information, such as the audio signal and speech turns, to improve translation using dialogue structure.

Table 2 :

 2 Automatic evaluation of baseline models.

			Ave. sent. Ave. sent. OpenSubs-mml	EUROPARL	BBT-train
	Corpus	# sents.	len. (en)	len. (fr)	% OOVs	% OOVs	% OOVs
	BBT-train	9,592	9.0	8.4	2.1		4.4	0
	BBT-trainm	7,462	9.0	8.5	15.4		23.0	0
	BBT-trainf	1,941	8.9	8.1	8.5		17.4	0
	BBT-dev	2,089	9.1	8.1	2.1		4.0	7.3
	BBT-test	1,941	9.2	8.2	1.9		4.0	7.2
	BBT-testm	1,438	9.4	8.4	2.3		4.1	8.0
	BBT-testf	354	8.8	8.0	1.1		4.0	4.7
	EUROPARL	1,969,197	27.1	30.0
	OPENSUBTITLES	27,737,442	9.4	8.9
	OpenSubs-mml	2,218,997	5.9	5.4
			Table 1: Corpus statistics
				BBT-dev	BBT-test
	Model data			BLEU	METEOR	BLEU	METEOR
	BBT-train			13.64	0.327	13.95	0.326
	EUROPARL			14.11	0.328	13.99	0.326
	OpenSubs-mml		23.20	0.419	23.69	0.432
	EUROPARL + BBT-train		16.86	0.362	16.82	0.368
	OpenSubs-mml+ BBT-train		23.39	0.419	24.09	0.434
	OpenSubs-mml+EUROPARL	24.55	0.430	24.74	0.441
	OpenSubs-mml+EUROPARL + BBT-train	24.64	0.428	24.56	0.437

Table 3 :

 3 Translation performance after adaptation of the OpenSubs-mml+EUROPARL model

			BBT-testm	BBT-testf
	Model adaptation Tuning data	BLEU	METEOR	BLEU	METEOR
		(i) Choice of the tuning set
	∅	all	23.91	0.434	25.16	0.450
	∅	male	24.09	0.438	25.72	0.450
	∅	female	23.67	0.431	25.22	0.446
		(ii) Addition of a gender-specific language model
	+LMm	all	24.17	0.436	24.80	0.447
	+LMf	all	23.35	0.430	24.13	0.443
	+LMm	male	23.92	0.435	25.39	0.448
	+LMf	female	23.97	0.444	26.25	0.459
		(iii) Addition of a gender-specific translation model
	+TMm	all	23.94	0.436	25.12	0.443
	+TMf	all	23.71	0.432	24.93	0.447
	+TMm	male	23.84	0.433	25.25	0.443
	+TMf	female	23.54	0.432	25.38	0.450
	(iv) Addition of a gender-specific language model and translation model
	+LMm+TMm	all	24.06	0.434	25.36	0.449
	+LMf+TMf	all	23.60	0.431	24.55	0.444
	+LMm+TMm	male	24.18	0.436	25.69	0.451
	+LMf+TMf	female	22.64	0.422	24.91	0.441

http://www.opensubtitles.org

Although not strictly spontaneous speech, subtitles offer an approximation of speech and dialogue, and despite often containing a number of non-literal translations, have the advantage of existing in large quantities for multiple language pairs.

This information is not available in the source sentence and so is only attainable through knowledge of speaker gender.

Due to copyright, the corpus cannot be distributed, but is reproducible using the plugin, once the DVD has been purchased.

Using the Tesseract software (Smith, 2007) and VobSub2SRT (https://github.com/ruediger/VobSub2SRT)

http://bigbangtrans.wordpress.com

Gender identification is a standard part of speaker diarisation systems (e.g.[START_REF] Barras | Multistage speaker diarization of broadcast news[END_REF].

We remove all episodes from The Big Bang Theory from this second corpus to avoid any bias.

For example, the model EUROPARL+BBT-train contains two phrase tables (with the either strategy implemented in Moses), one for each corpus, and two language models, one for each corpus. A single reordering model is used for each system, based on the largest corpora used for the system.

We even identify a case of reported speech uttered by a female speaker, in which the gender was erroneously corrected: "The man said 'I am a physicist"', translated as "L'homme a dit 'je suis physicienne"', with a feminine suffix -ienne.

We annotated all 114 of the female utterances that differed between the baseline and adapted model, and we randomly selected 200 of the 523 differing sentences for male speakers (of a total of 1,438 test sentences).