A Rona

M Grottadaurea

M Monti

C Airiau

T Gandhi

Generation of a Turbulent Boundary Layer Inflow for RANS Simulations

The generation of a fully turbulent boundary layer profile is investigated using analytical and numerical methods over the Reynolds number range 300 ≤ Re θ ≤ 31000. The predictions are validated against reference wind tunnel measurements with a zero streamwise pressure gradient. The analytical method is then tested for favourable pressure gradient by modelling the turbulent boundary layer approaching a two-dimensional potential sink. Both methods show a good predictive ability under a zero pressure gradient, with the numerical method providing a complete velocity profile through the laminar sub-layer down to the wall. This work is of practical interest to computational fluid dynamic practitioners for generating an equilibrium thick turbulent boundary layer at the computational domain inflow.

Nomenclature

I. Introduction

Computational fluid dynamic simulations of wall bounded flows, such as the flow over high-lift devices, ailerons, the elevators and the rudder, often use a turbulent boundary layer inflow to reduce the computational domain size with respect to a full wing, tailplane or fin simulation. The quality of the numerical predictions can be significantly affected by how well the boundary layer inflow is modelled. This paper compares the use of analytical correlations and of an auxiliary boundary layer numerical method to generate a turbulent boundary layer inflow for CFD over a wide Reynolds number range.

Computational fluid dynamic simulations of individual airframe components are commonly used to study the local aerodynamics in details. [START_REF] Gandhi | Injection parameters for an effective passive control of the cavity flow instability[END_REF][START_REF] Grottadaurea | Noise sources from a cylindrical cavity[END_REF][START_REF] Grottadaurea | The role of the inflow momentum thickness in subsonic cylindrical cavity noise generation[END_REF] This enables to achieve a sufficient level of spatial and temporal refinement around the specific components to model the onset of self-sustained oscillations, such as those in cavity flows, edge tones and other fluid-resonant geometries. These flow instabilities contribute to airframe noise and a good quality inflow prediction is very important to achieve quantitative predictions of the radiated noise pattern. For instance, in a cavity, the inflow momentum thickness has a direct influence on acoustic mode selection. [START_REF] Rowley | On Self-sustained Oscillations in Two-dimensional Compressible Flow Over Rectangular Cavities[END_REF] Where the inflow features a fully developed turbulent boundary layer, an analytical profile for the mean velocity can be imposed, derived from the integral boundary layer parameters as determined from either a larger-scale numerical simulation or from experiment. A common choice for specifying the boundary layer inflow in aerodynamics is by defining the inflow free-stream velocity, u e , temperature T e , pressure p e , boundary layer thickness δ, momentum thickness based Reynolds number Re θ , shape factor H, and the streamwise pressure gradient dp e /dx.

An alternative approach to define the mean velocity inflow is by using an auxiliary numerical simulation of the upstream boundary layer obtained, for instance, from running two-dimensional companion software by Wilcox. 5 This paper presents and validates one analytical and one numerical approach for generating a turbulent boundary layer inflow in CFD. The analytical method is a variant of the defect law by Coles, [START_REF] Coles | The law of the wake in the turbulent boundary layer[END_REF] while the numerical method is derived from matched asymptotic expansions. [START_REF] Cousteix | Asymptotic analysis and boundary layers[END_REF] The analytical approach is then applied to a boundary layer with favourable pressure gradient, in which a two-dimensional potential sink is used to generate the outer flow.

Section II details the analytical method used to generate the outer layer velocity profile in a turbulent boundary layer. Section III details the numerical method based on the equilibrium boundary layer model.

Section IV validates both methods using zero pressure gradient velocity data over the Reynolds number range 300 ≤ Re θ ≤ 31000. Section V presents a two-dimensional sink flow and extends the validation to the favourable pressure gradient boundary layer approaching the sink.

II. Analytical method

To describe the mean velocity profile in a turbulent boundary layer, similarity solutions are sought in the inner and the outer regions. In the inner region, the mean streamwise velocity u scales with the wall friction velocity u τ and with the viscous length scale l = ν/u τ , so that

u u τ = f y + (1)
where y + = yu τ /ν is the non-dimensional wall-normal distance. In outer region, the velocity profile is described by the velocity defect law

u ∞ -u u τ = f (η) (2)
where η = y/δ is the non-dimensional wall-normal distance, u e is the free-stream velocity, ν is the kinematic viscosity, y is the wall-normal distance and δ is the boundary layer thickness, which is taken as the wallnormal distance at which u = u e .

Based on the existence of an overlap region between the inner and the outer regions, Coles [START_REF] Coles | The law of the wake in the turbulent boundary layer[END_REF] proposed the following additive law of the wall and law of the wake in non-dimensional form:

u + = 1 κ ln y + + B + Π κ f (η) f (η) = 1 -cos (πη) (3)
where u + = u/u τ is the normalized streamwise velocity, Π is the wake parameter, κ the von Kármán constant, and B the logarithmic law constant.

Coles [START_REF] Coles | The law of the wake in the turbulent boundary layer[END_REF] determined the wake parameter as

Π = κ/2 u + e -κ -1 ln Re τ -B (4)
where Re τ = δu τ /ν is the boundary layer Reynolds number and u + e = u e /u τ is the normalized free-stream velocity.

Let

f (η) = A 1 η 2 + A 2 η 3 (5)
be a cubic polynomial approximation to f (η) in eq. 3. Substituting the boundary conditions

u| y=δ = u e (6)
and ∂u ∂y y=δ = 0 (7) in eq. 3, with f (η) from eq. 5, gives

A 1 = 6 [1 + 1/(6Π)] and A 2 = -4 [1 + 1/(4Π)],
with Π defined by eq. 4.

The law of the wake of eq. 3 then becomes

u + =
Log-law of the wall

1 κ ln y + + B + 1 k η 2 (1 -η) Pure wall flow +2 Π κ η 2 (3 -2η)
Pure wake component (8) Equation 8 is validated over a relatively wide range of momentum thickness based Reynolds number Re θ = u e θ/ν in section IV. To evaluate eq. 8, the authors take κ = 0.41 and B = 5.0, as proposed by Coles. 6

III. Successive complementary expansion method

The successive complemetary expansion method consists in seeking contiguous asymptotic matches between the inner and the outer regions of an incompressible turbulent boundary layer. This approach is detailed in Cousteix & Mauss 7 and just the key steps are reproduced in this paper that support the authors' application to turbulent boundary layers.

III.A. Mixing length model

Across the boundary layer, the local shear stress

τ = µ ∂u ∂y -ρu ′ v ′ = τ l + τ t (9)
where u ′ and v ′ are the time-dependent fluctuations of the streamwise and flow-normal velocity components and are unknown. To avoid having to resolve these unknowns, the Reynolds shear stress τ t is evaluated using Prandtl's mixing length model, [START_REF] Prandtl | Bericht über Untersuchungen zur ausgebildeten Turbulenz[END_REF] with the Van Driest 9 near-wall damping correction F. This gives

τ t = ρ F2 ℓ 2 ∂u ∂y ∂u ∂y (10)
where F = 1 -exp (-y + /26).

In the inner region, ℓ = κy, while in the outer region, ℓ/δ → 0.085 as y → δ. These two trends can be merged analytically into a single distribution for the mixing length ℓ across the full boundary layer by the use of a blending function. Michel et al. [START_REF] Michel | Application d'un schéma de longueur de mélange à l'étude des couches limites turbulentes d'équilibre[END_REF] used the blending function

ℓ(η) = δc ℓ tanh( κη c ℓ ) (11)
with c ℓ = 0.085 and κ = 0.41. The authors propose an alternative blending function that is shown in section IV to give an improved prediction of the turbulent shear stress profile at the interface between the inner and the outer layer, at low Reynolds numbers Re τ . This is

ℓ(η) = δ κη [1 + (κη/c ℓ ) n ] 1/n (12)
For 2.6 < n < 2.7, the ℓ(η) profile from equation 12 almost matches that from equation 11.

III.B. Inner region velocity profile

Normalising the local shear stress τ in eq. 10 by ρu 2 τ and assuming a monotonic velocity profile gives

τ τ w = ∂u + ∂y + + ℓ +2 F2 ∂u + ∂y + 2 (13)
In the limit y + → 0, τ → τ w and eq. 13 becomes

1 = ∂u + ∂y + + ℓ +2 F2 ∂u + ∂y + 2 (14)
where ℓ + = ℓu τ /ν. Equation 14 is a quadratic in ∂u + /∂y + with root 7

∂u + ∂y + = 2 1 + 1 + 4 ℓ + (y + ) F (y + ) 2 (15)
Integrating equation 15 with respect to y + with the boundary condition u + (x, 0) = 0 gives the inner layer tangential velocity profile that asymptotes to the log-law of the wall in equation 8 for y + → ∞.

III.C. Outer region velocity profile

In an equilibrium turbulent boundary layer, the similarity solution for the outer layer can be expressed in terms of the velocity defect F ′ (η) = u + e -u + . Expressing τ /τ w as a function of F and η gives 7

τ τ w = 1 - F F 1 + 1 F 1 + 2β ηF ′ (16)
where

F = η 0 F ′ (ξ) dξ; F 1 = F (1) ; β = - δ u τ du e dx (17)
In the outer region, the Reynolds stress component is dominant over the laminar shear stress, so τ ≃ τ t . From eq. 10, noting that the van Driest damping constant F → 1 at y + ≥ 100, τ /τ w = (ℓ/δ) 2 F ′′2 , where F ′′ = dF ′ /dη. Substituting for τ /τ w in eq. 16, the similarity solution for the outer region becomes

ℓ δ 2 F ′′2 = 1 - F F 1 + 1 F 1 + 2β ηF ′ (18)

III.D. Asymptotic matching of the inner and outer profiles

A matching condition is sought for the velocity profiles of the inner and outer regions, equations 15 and 18. This is obtained from standard asymptotic analysis 7 by considering eq. 15 in the limit y + → ∞ and eq. 18 in the limit η → 0 that give respectively 7

u + = κ -1 ln y + + C (19)
u + e -u + = -κ -1 ln η + D v (20)
Adding eq. 19 to eq. 20 gives 7

u + e = κ -1 ln

u τ δ ν + C + D v (21)
Equation 21 can be re-cast as function of the wall skin friction coefficient C f = τ w / 0.5ρu 2 e that is imposed as equal in the inner and outer regions and provides the matching criterion for the two profiles

2 C f = κ -1 ln u τ δ ν + C + D v (22)

III.E. Numerical implementation

Expliciting the outer region velocity profile poses several challanges. Equation 18 is non-linear and is illdefined in at the upper boundary layer limit, at η → 1, where F ′′ → 0, and at the lower boundary layer limit, at η → 0, where ℓ/δ → 0 and F ′′ → ∞. To solve the problem, auxiliary approximate solutions are imposed on the floor of the laminar sub-layer and at the edge of the boundary layer, as shown in figure 1 so that the edges of the inner and of the outer regions are modelled analytically while the overlap region is resolved numerically. Let f (η) = F (η) /F (1). On the floor of the laminar sub-layer, imposing η = 0 and ℓ = κy, as in section III.A, eq. 18 becomes

[κηF 1 f ′′ (η)] 2 = 1 -f (η) + (1 + 2βF 1 ) ηf ′ (η) (23)
with the boundary condition f (0) = 0. Let β = 2βF 1 . In a zero pressure gradient boundary layer, β = 0 by eq. 17, for which eq. 23 has the explicit solution

f (η) = η 2 4α 2 - η ln η α + A η; f ′ (η) = η 2α 2 - 1 + log η α + A; f ′′ (η) = 1 2α 2 - 1 αη with α = F 1 κ.
The integration constant A is determined by evaluating f ′ (η) at η = ǫ 0 on the floor of the laminar sub-layer. In a non-zero pressure gradient boundary layer, βηf ′ → 0 as η → 0, so the zero pressure gradient profile is used on the floor of the laminar sub-layer.

d (η) η = 1 -ǫ 1 η = 1
numerical integration of eq. ( 13) numerical integration of eq. ( 14) At the edge of the boundary layer, at η = 1, eq. 18 becomes

y + 0 = ǫ 0 × Re τ
[ℓ 1 F 1 f ′′ (η)] 2 = 1 -f (η) + 1 + β ηf ′ (η) (24)
with the boundary conditions f (1) = 1, f ′ (1) = 0, f ′′ (1) = 0 and ℓ 1 evaluated from eq. 12 at η = 1.

Cousteix [START_REF] Cousteix | Outer boundary layer self-similar solution[END_REF] proposed the solution for eq. 24

f (η) = 1 - (1 -η) 3 3 ; f ′ (η) = (1 -η) 2 ; f ′′ (η) = -2 + 2η (25)
for β = 0, that has the attractive property of being independent from F 1 and ℓ 1 and is the solution used in this work. The same solution is used for β = 0, as βηf ′ (η) = 0 by the boundary condition f ′ (1) = 0 in eq. 24.

IV. Zero pressure gradient boundary layers

The analytical and numerical methods for predicting a boundary layer mean turbulent velocity profile are tested against a range of streamwise velocity measurements from zero pressure gradient boundary layers 12-15 over the range 300 ≤ Re θ ≤ 31000. Table 1 lists the values of u + e , Re τ and Π at each Re θ of the experimental velocity traverse records. [12][START_REF] Erm | Low-Reynolds-number turbulent boundary layers[END_REF][START_REF] De Graaff | Reynolds-number scaling of the flat-plate turbulent boundary layer[END_REF][START_REF] Österlund | Experimental studies of zero pressure-gradient turbulent boundary layer flow[END_REF] The values of u + e and Re τ are the ones reported in experiment 12-15 while Π has been obtained by fitting eq. 8 using the least squares fit. The normalized mean streamwise velocity u + is plotted against the normalized wall-normal distance y + in figure 8 for different Reynolds numbers. The symbols used in figure 8 are measured values 12-15 at different Re θ , labelled as in table 1. The continuous lines show the fitted analytical profiles for the outer layer. For clarity, an incremental shift of u + = 2.5 is applied to all curves. The three 0 labels on the vertical axis of figure 8 correspond to Re θ = 300, Re θ = 5200, and Re θ = 31000 respectively. The quality of the predictions is quantified by evaluating the mean square percentage error ǫ for each profile

Re

ǫ = 1 N N i=1 u + a -u + e u + e 2 (26)
where u + a is the predicted value and u + e is the corresponding experimental value for a given y + i in a discretized velocity profile of N points. The ǫ obtained at different Re θ with u + a evaluated from equation 8 is reported in table 1. The maximum ǫ is 2.05% at Re θ = 31000. Such error enables the use of eq. 8 to predict the mean streamwise velocity of boundary layers in many common engineering applications, where an error margin of 5% is often acceptable. The experimental data seem to be randomly distributed about the fitted curve with no underlying trend, suggesting that the curve fit has captured most of the u + dependence on δ, u e , u τ , and Re θ .

Figure 3 compares velocity profiles obtained using the successive complementary expansion method of section III with the same experimental data of figure 2. n = 4 was used for the numerical prediction of the mixing length in eq. 12. The symbols used in figure 8 are measured values 12-15 at different Re θ , labelled as in table 1. The continuous lines show the normalized numerical velocity profiles. For clarity, the same incremental shift of u + = 2.5 as in figure 2 is applied to all curves. The origin of the ordinate of figure 3 refers to the Re θ = 300 profile. Figure 3 shows that the complementary expansion method of section III produces a full velocity profile down to the wall. In the outer layer, the complementary expansion method captures the Reynolds number dependent transition between the log-law and the constant free-stream velocity for most of the curves. The free-stream velocity at Re θ = 22845, 12663 and 3654 appear to be under-predicted. This is confirmed by the corresponding numerical mean square percentage error, ǫ num , which is computed by evaluating u + a in eq. 26 using the successive complementary expansion method. Specifically, the ǫ num at Re θ = 22845, 12663 and 3654 are higher than for some of the other Reynolds numbers, due to the difference in the normalized free-stream velocity between experiment and prediction. Whereas, in general, the error from the numerical velocity profile is higher than that from the analytical profile, it is within the range for which the predictions can be used for engineering accurate predictions. The difference between the normalized free-stream velocity from experiment and from the successive complementary expansion method is further investigated in figure 4, where the outer layer portion of the predicted velocity profile for Re θ = 22845 is re-plotted on a larger scale. The continuous black line is the numerical prediction obtained by matching the experimental value of Re θ in the matched complementary expansion, the red dash-dot line is obtained by matching the experimental value of Re τ , while the dashed blue line shows the predicted profile with a matched normalized free-stream velocity u + e . Matching the experimental Reynolds numbers seems to give similar profiles irrespective of whether the target Reynolds number is defined with respect to the momentum thickness, Re θ , or the friction velocity, Re τ . Fitting the outer profile by imposing the normalized free-stream velocity u + e seems to over-predict the boundary layer thickness, leading to a coarser fit with experiment compared to the numerical predictions obtained by matching the profile Reynolds number.

Figure 5(a) compares the mixing length distribution across a zero-pressure gradient boundary layer with ℓ (η) obtained from measurements at Re τ = 1540 by Klebanoff,[START_REF] Klebanoff | Characteristics of turbulence in a boundary layer with zero pressure gradient[END_REF] reported in Hinze. [START_REF] Hinze | Turbulence[END_REF] The ℓ distribution from equation 11 is shown by the continuous line while the dashed line shows the distribution from equation 12 with n = 4. At these conditions, there appears to be a good improvement in the predicted mixing length using the new formulation. No effort has been made to further optimize n ∈ ℜ by adding decimal digits. [START_REF] Michel | Application d'un schéma de longueur de mélange à l'étude des couches limites turbulentes d'équilibre[END_REF] eq. 11, under-predicts the eddy viscosity, as shown by the continuous line, whereas a better fit is achieved using eq. 12. As a numerical experiment, the target Reynolds number in the successive complementary expansion method was varied over the range 1000 ≤ Re τ ≤ 2775 and was found to have very little effect on the predicted normalized ν t , which is also the trend in experiment. [START_REF] Klebanoff | Characteristics of turbulence in a boundary layer with zero pressure gradient[END_REF][START_REF] Toensend | The structure of the turbulent boundary layer[END_REF] This paper has not attempted to predict the time-averaged velocity profiles of boundary layers at Re τ < 300 using the matched complementary expansion method. In this method, u + e is obtained by matching the outer layer velocity profile to the inner layer velocity profile in the logarithmic layer. When Re τ < 140, an overlap region in the form of a logarithmic layer is no longer present, which prevents the method form evaluating u + e . Here the matched complementary expansion method in its present formulation has reached its Re τ applicability limit. Consider a two-dimensional potential sink of strength Q located at the trailing edge of a two-dimensional flat plate, as sketched in figure 6.

V. Favourable pressure gradient boundary layers

The sink induces a streamwise velocity u 0 at the leading edge of the plate, at x = 0, from which an incompressible turbulent boundary layer develops along the plate length L.

It is assumed that the favourable pressure gradient induced by the potential sink of strength Q does not re-laminarize the boundary layer that remains turbulent over the full length L of the plate. Neglecting boundary layer displacement thickness effects, from mass conservation, the local time-averaged free-stream velocity u e is given by

u e u 0 = 1 1 -x/L . ( 27
)
The sink flow is uniquely characterized by the positive constant acceleration parameter

K = ν u 2 e du e dx = ν U 0 L . ( 28
)
The sink flow is a smooth wall boundary layer that satisfies the conditions for "precise equilibrium", [START_REF] Townsend | Equilibrium layers and wall turbulence[END_REF][START_REF] Rotta | Turbulent boundary layers in incompressible flow[END_REF][START_REF] Jones | Evolution and structure of sink-flow turbulent boundary layers[END_REF] in that the mean defect velocity profile and the Reynolds stress profile are invariant with the streamwise coordinate x. Coles 6 proposed that the sink flow at equilibrium gives a pure wall flow velocity profile, eq. 8, in which Π = 0. He provided several arguments to support his hypothesis but no rigorous proof was given that Π = 0 corresponds to sink flow at equilibrium. This work is presented on the premise that Coles' hypothesis is acceptable.

Perry et al. [START_REF] Perry | Wall turbulence closure based on classical similarity laws and the attached eddy hypothesis[END_REF] derived a formula for the total shear stress distribution along the flow-normal direction by combining the law of the wake with the mean continuity and mean momentum equations:

τ τ 0 = f 1 η, Π, u + e + g 1 η, Π, u + e β c , (29)
where β c = (δ * /τ 0 ) dΠ/dx is the Clauser parameter. [START_REF] Clauser | The turbulent boundary layer[END_REF] The explicit forms of f 1 , g 1 and g 2 are given in Perry et al. [START_REF] Perry | Wall turbulence closure based on classical similarity laws and the attached eddy hypothesis[END_REF] and depend only on the law of the wake. From the momentum integral equation and using the momentum and the displacement thicknesses, that depend on the wake function, the evolution equations are obtained

F 1 u + e 2 E (Π) exp κu + e K + β c = 0 (30) E (Π) exp κu + e u + e λ du + e dR x = R Π, u + e , β c (31)
where λ = u e /u 0 , R x = xu 0 /ν, E (Π) and R (Π, u + e , β c ) are detailed in Perry et al. [START_REF] Perry | Wall turbulence closure based on classical similarity laws and the attached eddy hypothesis[END_REF] Perry et al. [START_REF] Perry | Wall turbulence closure based on classical similarity laws and the attached eddy hypothesis[END_REF] matched the shear stress distribution along η = 0.4 of a quasi-equilibrium boundary layer sink flow to obtain

β = -f 1 (0.4, Π, u + e ) + f 1 (0.4, Π, ∞) g 1 0.4, Π, u + e + g 1 (0.4, Π, ∞) g 1 0.4, Π, u + e β ae (Π) , (32)
where β ae (Π) is the limit value of eq. 32 for u + e → ∞. [START_REF] Perry | A new look at some closure problems of turbulent boundary layers[END_REF] Green et al. [START_REF] Green | Prediction of turbulent boundary layers and wakes in compressible flow by lag-entrainment method[END_REF] proposed:

β ae = 0.03 C 2 (Π) F 1 2 -1.25, (33)
where C 2 (Π) = 1 0 F ′2 (η) dη. Jones et al. [START_REF] Jones | Evolution and structure of sink-flow turbulent boundary layers[END_REF] proposed an empiric formula for β ae ,

β ae = -0.5 + 1.38Π + 0.13Π 2 , (34)
based on curve fitting over the range 0 ≤ Π < 0.4. Equation 30 can be solved numerically to obtain Π (u + e ). Substituting this relation into eq. 31 and setting the initial condition u + 0 = 19 at x = 0, that corresponds to a freshly tripped turbulent boundary layer, eq. 31 can be solved to give

x/L = 1 -exp -K u + e u + 0 H u + e du + e , (35)
H u + e = u + e E [Π (u + e )] exp (κu + e ) R Π u + e , u + e , β c Π u + e , u + e . (36)
Perry et al. [START_REF] Perry | On the streamwise evolution of turbulent boundary layers in arbitrary pressure gradients[END_REF] generalized eq. 35 to include the effect of the wake strength gradient for non-equilibrium boundary layers by solving an ordinary system of differential equations.

Equation 35 describes implicitly the streamwise variation of the wall shear stress along the flat plate. This relationship is used in subsection V.B to compare the streamwise variation of the normalized shear stress S and of other boundary layer integral parameters between theory and experiment.

V.B. Sink flow results

In this subsection, the law of the wake of eq. 8 is used to predict the streamwise variation of the integral parameters characterizing the sink flow boundary layer of subsection V.A. This exercise follows that of Perry et al., [START_REF] Perry | Wall turbulence closure based on classical similarity laws and the attached eddy hypothesis[END_REF] where the laws of the wake of Coles 6 and Lewkowicz [START_REF] Lewkowicz | An improved universal wake function for turbulent boundary layers and some of its consequences[END_REF] were used. The aim is to show that there is some improvement in the prediction of these non-dimensional parameters by using eq. 8 in place of the laws of the wake in the literature [START_REF] Coles | The law of the wake in the turbulent boundary layer[END_REF][START_REF] Lewkowicz | An improved universal wake function for turbulent boundary layers and some of its consequences[END_REF] and that the analytical method presented herein is performing at least as well as the literature benchmark in predicting a pure wake flow.

Figures 7(a) and 7(b) show the changes in the non-dimensional parameters Π and u + e in the streamwise direction, as determined by solving eq. 35 using eqs. 33 and 8 (black curves) and eqs. 33 and the law of the wake by Lewkowicz [START_REF] Lewkowicz | An improved universal wake function for turbulent boundary layers and some of its consequences[END_REF] (red curves). The starting condition is u + 0 = 19 at x = 0 that corresponds to a turbulent boundary layer tripped at the leading edge, as sketched in figure 6. The numerical results of eq. 35 are obtained at the three different values of the acceleration parameter, K = 2.7 × 10 -6 (solid curve), K = 3.56 × 10 -6 (dash-dot curve), and K = 5.26 × 10 -6 (dashed curve). In figs. 7(a) and 7(b), these predictions are compared against experimental values from Jones et al. [START_REF] Jones | Evolution and structure of sink-flow turbulent boundary layers[END_REF] at K = 2.7 × 10 -6 ( * ), at K = 3.56 × 10 -6 (+), and at K = 5.26 × 10 -6 (△). Using Lewkowicz' law of the wall, eq. 35 results in a distribution of Π (red curves) significantly higher than in experiment (symbols). The over-prediction from using the law of the wall of Lewkowicz (red curves) is partially recovered by using eq. 8 that introduces a vertical shift of Π ≈ -1/6 in the Π vs u + e plot. This results in the black curves that are identical in shape to the red curves but are closer to the experimental values, due to the Π ≈ -1/6 vertical shift. Figure 7(b) shows the variation of the wake parameter Π in the streamwise direction. The physical significance of the wake parameter is to allow for the presence of a wake region at the top of a turbulent boundary layer in which the velocity profile departs from being a pure wall flow. The acceleration due to the potential sink is such to stop the growth of the wake region, so that close to the sink the wake region thickness approaches zero. This implies that the proper asymptotic value of Π close to the potential sink is 0. Using Lewkowicz' law of the wall and eq. 33 results in Π ≈ 1/6 as x/L → 1, as shown by the red curves in figure 7(b). The correct asymptotic behavior of Π → 0 as x/L → 1 is recovered when eq. 8 and eq. 33 are used, as shown by the black curves in fig. 7(b). This indicates that the new law of the wake is an improvement over Lewkowicz' formulation in predicting the asymptotic wake region of a turbulent boundary layer with a strong favourable pressure gradient.

The variation of Re θ and of u 0 δ/ν with x/L using eq. 33 and eq. 8 is identical to the one given in Jones et al., [START_REF] Jones | Evolution and structure of sink-flow turbulent boundary layers[END_REF] where the law of the wake by Lewkowicz 27 was used, and is therefore not reported here.

Figures 8(a) to 9(b) show the streamwise variation of the non-dimensional parameters obtained using the empirical formula of eq. 34 in place of eq. 33. In these figures, the curves show predictions from solving eq. 35. The red curves are obtained using the law of the wake by Lewkowicz [START_REF] Lewkowicz | An improved universal wake function for turbulent boundary layers and some of its consequences[END_REF] and 34 and the black curves are obtained using eqs. 8 and 34. The same line patterns (solid, dashed, dash-dot) as in fig. 7(a) are used to identify respectively the same three values of the acceleration parameter K. The symbols are experimental values from Jones et al. [START_REF] Jones | Evolution and structure of sink-flow turbulent boundary layers[END_REF] and are in the same notation as in fig. 7(a).

Figure 8(a) shows that, at the lowest value of the acceleration parameter K = 2.7 × 10 -6 , there is a significant improvement in the prediction of the variation of Π with u + e . The black curve fits through the cloud of experimental data ( * ) better than the red curve, which under-estimates Π at this value of K. At the higher acceleration parameter K = 3.56×10 -6 , the dash-dot curves from the law of the wake by Lewkowicz 27 and 8 lie respectively below and above the experimental data (+), with neither equation giving a significantly better prediction than the other. At the highest value of the acceleration parameter K = 5.26 × 10 -6 , neither equation gives a good fit to the experimental data, although the red dashed line from the law of the wake by Lewkowicz 27 is the closest.

Figure 8(b) shows the streamwise variation of the wake parameter Π predicted using eq. 34. While in eq. 33 β ae depends directly on the boundary layer shape η, eq. 34 is an empirical regression with constant coefficients, invariant on the profile shape. Therefore the streamwise Π distribution in figure 8(b) is also insensitive to changes in the law of the wake, resulting in predictions that essentially overlap for all values of K. The predictions are an improved fit to the experimental data with respect to figure 7(b), which reflects the regression used to obtain eq. 34. The asymptotic trend of Π → 0 as x/L → 1 is captured less well than with eqs. 8 and 33, black curves in figure 7(b), possibly due to the lack of experimental values at x/L → 1 on which eq. 34 is regressed.

Figure 9(a) shows the streamwise variation of the normalized boundary layer thickness at different values of the acceleration parameter K. The symbols are experimental values from Jones et al. [START_REF] Jones | Evolution and structure of sink-flow turbulent boundary layers[END_REF] taken at the same values of K as in figure 8(b). These data are labelled as in fig. 8(a). Predictions of δu 0 /ν are obtained using eq. 8 (black curves) and using the law of the wake by Lewkowicz 27 (red curves). The best match between prediction and experiment is obtained by using eq. 8 at all K. This is because in deriving eq. 8 the height of the boundary layer δ is used as one of the closure parameters, whereas the law of the wake by Lewkowicz 27 satisfies eq. 6 but not eq. 7, leading to the coarser match in the streamwise variation of the normalized boundary layer thickness shown in figure 9(a).

Figure 9(b) shows the streamwise variation of the momentum thickness based Reynolds number approaching the sink located at x = L. The same notation as fig. 9(a) is used. Re θ grows monotonically in the downstream direction, with a reducing rate at increasing values of the acceleration parameter K. Approaching the sink, the flow acceleration reduces the growth rate to zero, so that Re θ asymptotes to a constant value that is K dependent. The predictions of Re θ appear to be weakly dependent upon whether the law of the wake by Lewkowicz [START_REF] Lewkowicz | An improved universal wake function for turbulent boundary layers and some of its consequences[END_REF] or eq. 8 are used for their estimate. The relative insensitivity of the momentum thickness Reynolds number on the wake function agrees with the findings from using eq. 33 and eq. 8, the predictions from which are identical to those given in Jones et al. [START_REF] Jones | Evolution and structure of sink-flow turbulent boundary layers[END_REF] Comparative predictions of the boundary layer parameters approaching the sink flow from the matched complementary expansion method are not given due to the difficulty of finding a solution at the same values of the acceleration parameter K as in experiment. The acceleration reduces the thickness of the outer layer, so that the method fails to find a match between the inner and the outer layer velocity profile that is continuous in u + and in du + /dy + across the interface. This outcome is in-line with the sink flow satisfying the conditions for "precise equilibrium" and giving a pure wall flow velocity profile, which is discontinuous at η = 1. A preliminary analysis of the parameter space for K for which a solution by the matched complementary expansion is available suggests that the method works for K < 10 -6 .

VI. Conclusions

Numerical and analytical methods for obtaining the time-mean velocity profiles of a turbulent boundary layer are presented and validated against experimental data.

The analytical method is an extension to the law of the wake by Coles 6 that matches both the free stream velocity and the velocity gradient at the boundary layer edge. The method is shown to predict the outer region of turbulent boundary layers rather well for zero streamwise pressure gradient test cases over the Reynolds number range 300 ≤ Re θ ≤ 31000, with a maximum mean square percentage error of 2.05%.

The authors propose a modification to the successive complementary expansion numerical method in Cousteix & Mauss, [START_REF] Cousteix | Asymptotic analysis and boundary layers[END_REF] with a new blending function for the mixing length in the outer region. Comparison against experimental data shows that the new blending function improves the prediction of the mixing length and of the eddy viscosity in outer region of a zero pressure gradient boundary layer. The new method is validated against experimental velocity profile data over the Reynolds number range 300 ≤ Re θ ≤ 31000 under zero streamwise pressure gradient and found to achieve engineering accurate predictions. The new blending function introduces an additional adjustable parameter n ∈ ℜ in the model that can undergo a more extensive calibration over a wider experimental dataset to further improve the predictions.

A sink flow test case was used to study the change in the wake region thickness in an accelerating flow. By using the new law of the wake, the correct asymptotic behavior for the wake parameter Π is recovered, namely Π → 0 at x/L → 1, when the streamwise variation of Π is evaluated from the integral length scales F 1 and C 2 of the defect law under quasi-equilibrium conditions.

Using a parabolic distribution for the Clauser parameter at equilibrium, β ae , independent from F 1 and C 2 , reduces the sensitivity of the predictions on the choice of the wake function. The new law of the wake improves to some extent the predictions of the non-dimensional boundary layer thickness distribution.

  ln η u = u e η = ǫ 0 ln y + analytical solution u

Figure 1 .

 1 Figure 1. Boundary layer decks.

Figure 2 .

 2 Figure 2. Turbulent boundary layer profiles fitted to eq. 8. Symbols as in table 1.

Figure 3 .

 3 Figure 3. Turbulent boundary layer profiles fitted by the complementary expansion method. Symbols as in table 1.

Figure 4 .

 4 Figure 4. Outer layer profile fitted by the complementary expansion method. Re θ = 22845. (+) experiment, (-) successive complementary expansion method.

Figure 5 (

 5 b) shows the profile of the normalized eddy viscosity ν t / (u τ F 1 δ) across the same zero pressure gradient boundary layer, where ν t = F2 ℓ |∂u/∂y|. The symbols are from the same experiment 16 as in figure 5(a) (open circles) to which further measurements by Townsend 18 at Re τ = 2775 have been added (open squares). Using the mixing length model of Michael et al.,

  Mixing length ℓ versus normalized distance from the wall η at Reτ = 1540. Normalized eddy viscosity νt uτ δ F 1 versus normalized distance from the wall η.

Figure 5 .

 5 Figure 5. Turbulence model variables. (•) experiment 16 at Reτ = 1540, (2) experiment 18 at Reτ = 2775, (--) matched complementary expansion at Reτ = 1000 with F1 = 3.1479 from eq. 11, (-) matched complementary expansion at Reτ = 1000 with F1 = 3.1044 from eq. 12.

Figure 6 .

 6 Figure 6. Sink flow.

  Variation of the wake parameter Π with the normalized free-stream velocity u + e . Streamwise variation of the wake parameter Π.

Figure 7 .

 7 Figure 7. Variation of the wake parameter Π in a sink flow. βae from eq. 33.

  Variation of the wake parameter Π with the normalized free-stream velocity Lewkowicz. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0Streamwise variation of the wake parameter Π.

Figure 8 .

 8 Figure 8. Variation of the wake parameter Π in a sink flow. βae from eq. 34.

Figure 9 .

 9 Figure 9. Streamwise variation of sink flow parameters. βae from eq. 34.

Table 1 .

 1 Experimental velocity profiles.

	θ	Re τ	u + e	Π	100 × ǫ Symbol (Re τ ) num (u + e ) num 100 × ǫ num
	300	145	18.25 0.228	1.33	•	142	18.54	2.12
	697	335	20.25 0.219	1.35	*	315	20.77	3.31
	1003	460	21.5 0.317	1.78	△	446	21.66	2.39
	1430	640	22.4 0.336	1.38	•	627	22.51	2.77
	2900	1192 24.33 0.421	1.02	⊳	1240	24.17	2.48
	3654	1365 25.38 0.568	0.72	×	1551	24.71	2.44
	5200	2000	26	0.505	1.62	⊲	2185	25.54	2.38
	12633 4436 28.62 0.643	0.71		5188	27.65	2.51
	13000 4770	28	0.480	0.99	3	5335	27.72	1.84
	22845 8000 30.15 0.662	1.01	+	9258	29.06	2.34
	31000 13030	30	0.388	2.05	⋆	12845	29.79	1.86

Acknowledgments

This research project has been supported by a Marie Curie Research Training Fellowship of the European Community's Sixth Framework Programme under contract MEST CT 2005 020301. This paper was written with the support of the Invited Professorship scheme of the Université Paul Sabatier, Toulouse.