APPLICATION OF THE DEFECT FORMULATION TO THE INCOMPRESSIBLE TURBULENT BOUNDARY LAYER

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1 APPLICATION OF THE DEFECT FORMULATION TO THE INCOMPRESSIBLE TURBULENT BOUNDARY LAYER O. ROUZAUD ONERA OAt1 29 avenue de la Division Leclerc - B.P CHATILLON Cedex - France AND B. AUPOIX AND J.-PH. BRAZIER CERT-ONERA DERAT 2 avenue E. Belin - B.P TOULOUSE Cedex - France 1. Introduction The defect formulation consists of working inside the boundary layer not on the physical variables but on the differences of those variables with the inviscid flow terms. The combination of this approach with the matched asymptotic expansions method is proved to be very satisfactory in the case of the interaction of an inviscid rotational flow with a laminar boundary layer. The predictions of the wall values and the boundary layer profiles are generally improved when compared to the results of Van Dyke s higher order boundary layer theory (Brazier, 1990; Aupoix and al., 1992). Inthiswork,anextensionofthedefectformulationisproposedforanincompressible turbulent boundary layer developing on a flat wall (Rouzaud, 1994). The flow under consideration is two-dimensional and steady. Besides, the longitudinal pressure gradient is supposed moderate which means that the phenomenon of separation or the effects of a strong pressure gradient are not taken into account. As the oncoming stream is rotational (fig.1), the main problem is to provide a correct matching between the boundary layer and the oncoming flow. In fact, the asymptotic analysis developed by Van Dyke, which is based upon the physical variables - called in this article the standard approach - can handle non-linear variations of the external flow with the height y only at second or higher order of approximation.

2 2 O. ROUZAUD ET AL. Y U(Y) External flow Boundary layer O( δ) Outer layer ^ O( δ) Inner layer L X Figure 1. Configuration flow Figure 2. Asymptotic structure Due to these considerations, the defect formulation is applied to the model of Yajnik (1970) and Mellor (1972). In the second section, the main hypothesis of the Yajnik-Mellor model and the asymptotic structure of the flow are recalled. Afterwards, the asymptotic analysis with the defect approach is carried out. In particular, the matching between the different regions is detailed and a comparison of the results with those of the Yajnik-Mellor model is given. In order to simplify the numerical study, the method of recombination of equations is also presented. The third section is devoted to a numerical comparison between the Prandtl turbulent solutions and the solutions obtained by the defect approach. In the fourth section, improvements of the previous results due to coupling methods for both standard and defect approaches are presented. 2. Application of the defect formulation 2.1. DIMENSIONLESS REYNOLDS EQUATIONS For the analysis, three reference values are introduced. They correspond to a velocity scale representative of the mean motion U, a length scale L and a turbulent velocity scale u t. The dimensionless Reynolds equations for incompressible two-dimensional flows are written as : u x + v = 0 u u x +v u = p x + x u v x +v v = p + x ( Γ xx + 1 ) u + ( Γ xy + 1 ) u Re x Re ( Γ xy + 1 ) v + ( Γ yy + 1 ) v Re x Re

3 DEFECT FORMULATION 3 R e = U L Reynolds number of the flow ν with Γ.. = u2 t < u.u. > U 2 u 2 Reynolds stresses t The ratio ε = ut U represents the turbulent intensity. It is assumed to be small compared to unity (ε 1). The asymptotic analysis is performed for high Reynolds number R e ASYMPTOTIC STRUCTURE OF THE BOUNDARY LAYER According to Yajnik and Mellor, the asymptotic model relies on three hypotheses. The first one establishes a relation between the time scale of the mean motion advection and the time scale of the turbulent diffusion. Let δ be the boundary layer thickness, the two time scales are assumed to be of the same order of magnitude and one obtains : δ L ε = u t δ u t U U L The second assumption is connected to the turbulent kinetic energy balance. Near the wall, in a region of thickness δ, it is known that the dissipation term ǫ, the viscous diffusion contribution D = (ν k ) and the turbulent production term P = < u v > u are equivalent. Such an hypothesis gives : δ L ν δ u t δ L = 1 εr e The above relation shows that the Reynolds number of the turbulence (Rt = ut δ ν ) is of order unity. For convenience, a small parameter ε is introduced as ε ε = δ. Finally, the third hypothesis is about the order of magnitude of the Reynolds stresses. Inside the boundary layer, the square of the turbulent velocity scale u t is representative of the Reynolds stresses scale. The asymptotic structure of the flow is then represented by an external flow and two regions located inside the boundary layer (fig.2). Inside the outer layer, the inertial and turbulent terms are predominant. Near the wall, viscous and turbulent contributions are the most important.

4 4 O. ROUZAUD ET AL EXTERNAL FLOW The variables depend on the (x, y) coordinates and the asymptotic expansions are : u = U 1 (x,y)+εu 2 (x,y)+ v = V 1 (x,y)+εv 2 (x,y)+ p = P 1 (x,y)+εp 2 (x,y)+ These are the expansions used in the Yajnik-Mellor model. As a matter of fact, the asymptotic equations of the first-order correspond to the Euler equations and are written as : U 1 x + V 1 = 0 U 1 U 1 x +V U 1 1 = P 1 (1) x U 1 V 1 x +V 1 V 1 = P 1 To the second-order, the system of equations represents the small perturbation equations of the inviscid flow : U 2 x + V 2 = 0 U 2 U 1 x +U U 1 2 x +V U 2 1 +V U 1 2 = P 2 (2) x U 1 V 2 x +U 2 V 1 x +V 1 V 2 +V 2 V 1 = P 2 No viscous terms are present for these orders of approximation. So, the external flow is regarded as inviscid OUTER LAYER In this region, the coordinates are (x,y = y δ ). The defect variables are introduced for the components of the velocity and the pressure. For example, the longitudinal velocity u is the sum of a defect variable u D (x,y) and of the inviscid component U(x, y). The expansions are: u = U(x,y)+u D (x,y) = U 1 (x,y)+εu 2 (x,y)+ +u 1 (x,y)+εu 2 (x,y)+ v = V(x,y) V(x,0)+v D (x,y) = V 1 (x,y) V 1 (x,0)+ +εv 1 (x,y)+ε 2 v 2 (x,y)+ p = P(x,y)+p D (x,y) = P 1 (x,y)+εp 2 (x,y)+ +p 1 (x,y)+εp 2 (x,y)+ Γ.. = ε 2 γ..1 (x,y)+

5 DEFECT FORMULATION 5 The normal velocity expansion allows to have an homogeneous condition for the defect term at the wall. The gauge of the term v 1 (x,y) ensures a nontrivial solution for the first-order continuity equation. On the other hand, the external terms V i (x,y) V i (x,0) and their longitudinal derivatives are supposed to be of order ε in the outer layer. They are written as V i (x,y) V i (x,0) = εv i (x,y). The asymptotic equations for the defect variables are obtained by introducing the previous expansions in the Reynolds equations and subtracting the inviscid equations of the external flow. At the, the system is : u 1 x + v 1 = 0 (u 1 +U 1 ) u 1 x +u 1 0 = p 1 U 1 x +(v 1 +V 1 ) u 1 V 1(x,0) U 1 = p 1 x (3) and at the second order, it can be written as : u 2 x + v 2 = 0 (u 1 +U 1 ) u 2 x +(u 2 +U 2 ) u 1 x +u 1 +(v 1 +V 1 ) u 2 +(v 2 +V 2 ) u 1 +v 1 U 1 V 1 (x,0) x V 1 (x,0) V 1 = p 2 U 2 x +u 2 U 1 x V 1(x,0) U 2 U 1 V 2(x,0) U 1 = p 2 x + γ xy1 (4) As in the standard approach, the inertial and turbulent terms are in balance, the Reynolds stresses only appearing at the second order INNER LAYER The inner coordinates are defined by (x,ŷ = y δ), and the expansions are: u = U 1 (x,y)+εu 2 (x,y)+ +û 1 (x,ŷ)+εû 2 (x,ŷ)+ v = V(x,y) V(x,0)+v D (x,ŷ) p = P(x,y)+p D (x,ŷ) Γ.. = ε 2 γ..1 (x,ŷ)+

6 6 O. ROUZAUD ET AL. Thegauge of the first-order variable v D is ε ε. Following thepreceding analysis, the terms V i (x,y) V i (x,0) are written as ε ε V i (x,y). The introduction of the expansions in the Reynolds equations lead at the to : û 1 x + v 1 ŷ = 0 0 = ( ) û1 (5) ŷ ŷ 0 = p 1 ŷ and at the second order to : û 2 x + v 2 ŷ = 0 0 = ( γ xy1 + û 2 ŷ ŷ 0 = p 2 ŷ ) (6) In this region, the main contributions are due to the viscous and turbulent terms MATCHING External flow / Outer layer Considering the expansions inside the outer layer, the limits of the defect variables as y tends towards infinity are given by : u D (x,y) = u U 0 v D (x,y) = v (V V(x,0)) V(x,0) as y p D (x,y) = p P 0 The first-order solution of the outer layer is described. According to the transverse momentum equation of the system (3), the defect pressure p 1 is not dependent on y. Considering the boundary conditions (7), the term p 1 is equal to zero, and the longitudinal velocity u 1 is also null. Due to the continuity equation of the system (3), the velocity v 1 is only dependent on the variable x. As there is a shift of gauges between the normal velocity expansions in the outer and inner layers, this term is also null, giving at first-order the solution u 1 v 1 p 1 0. Afterwards, the attention is focused on the inviscid flow. With the previous solution in the outer layer, the boundary conditions (7) and the expansions of the normal velocity in both zones prove that the normal velocities (7)

7 DEFECT FORMULATION 7 V 1 (x,0) and V 2 (x,0) of the external flow are equal to zero. As the boundary conditions of the system (2) are homogeneous, the inviscid solution (U 2,V 2,P 2 ) is uniformly null (Mellor, 1972). At last, the transverse momentum equation of the system (4) shows that the defect pressure p 2 is null. It means that, for the first two orders of approximation, the pressure p is equal to the local value of the inviscid pressure and may depend on the height y. It is a significant difference with the standard approach, because in this approach, the pressure is constant inside the boundary layer. To conclude with these matchings, it is interesting to build the firstorder solution inside the outer layer. For example, the longitudinal velocity is written as u = U 1 (x,y) + u 1 (x,y) = U 1 (x,y). By comparison with the standard approach where the local value of the longitudinal velocity is equal to the inviscid velocity U 1 (x,0) at the wall, the matching is more satisfactory. Even to the second order, the standard approach can only take into account a linear variation of the longitudinal velocity as the secondorder inviscid flow is null Outer layer / Inner layer The main results concern the pressure and the longitudinal velocity. For the pressure, the transverse momentum equations of the sytems (5) and (6) prove that the defect pressure is only dependent on the abscissa x. Considering the matching of the two layers, the defect pressure is assumed to be null inside the inner region. For the longitudinal velocity, the longitudinal momentum equation of the system (5) gives a linear variation of û 1 with the height ŷ. Due to the no-slip condition at the wall and the matching relation with the outer layer, the variable û 1 is equal to the opposite of the first-order inviscid velocity at the wall, U 1 (x,0). To the next order, we have to match the expansions : { ud (x,y) = εu 2 (x,y)+ u D (x,ŷ) = U 1 (x,0)+εû 2 (x,ŷ)+ As suggested by Mellor (1972), the problem of the shift of the gauges between the two expressions is solved by matching the normal derivatives of the velocity. In the present work, such a matching gives a relation between the defect velocities : y u D = ŷ û D (8) ŷ By integrating the previous relation, one obtains the behaviour of the defect velocity in the matching region. The defect velocity obeys a logarithmic behaviour and the expansions of the longitudinal velocity can be written

8 8 O. ROUZAUD ET AL. as : { u U1 (x,y)+ε(a(x)ln(y)+b(x))+ as y 0 u U 1 (x,y) U 1 (x,0)+ε(a(x)ln(ŷ)+c(x))+ as ŷ the parameters A(x), B(x), C(x) coming from the integration of the equation (8). In fact, the defect component acts upon the inviscid velocity as a perturbation velocity. These relations are different from those obtained by the standard approach. They provide a more general law of the wall as the dependency of the external flow with the y coordinate is taken into account RECOMBINATION OF THE EQUATIONS Before performing the numerical study, the recombination of the equations is introduced. The purpose of this method is to simplify the calculations by obtaining a global set of equations for the boundary layer. For the defect formulation, the method can be used either for the defect variables (u D,v D,p D ) or for the physical variables (u,v,p). In this article, we only present the second possibility. In a first step, the systems (3) and (4) of the inner layer are respectively added to the systems (1) and (2) of the external flow. The second step consists in applying the same process to the outer layer and the external flow. The next stage is the union of the two sets of equations to form a global system valid for the whole boundary layer. Once the physical variables (u, v, p) are reintroduced, each term belonging to one system or the other is retained. For example, the viscous stress only appears in the inner layer, but it must be kept as viscous effects are important in the boundary layer. At last, the global system is : u x + v = 0 u u x +v u = p x < u v > (u U) R e 2 +(v V) U p(x,y) = P(x,y) (9) At the wall, the no-slip condition is ensured and at the upper edge, the velocity u tends towards the inviscid velocity U(x,y) as y tends towards infinity. The system may be regarded as first-order significant because it contains turbulent terms. Applied to the standard approach of the Yajnik-Mellor model, the recombination provides the Prandtl turbulent equations with the usual boundary conditions.

9 DEFECT FORMULATION 9 3. Numerical study 3.1. TEST CASES The flow configuration is recalled on figure 1. The oncoming flow is supposed parallel, symmetrical with y. Its shear is either constant or dependent on the height y. The boundary layer develops on a flat wall. In the test case, the first-order inviscid flow is described by U 1 (x,y) = f(y),v 1 (x,y) = 0,P 1 (x,y) = C st. The results are presented for a linear (U = 1+5y) and a hyperbolic distribution (U = 1+625y 2 ) but other distributions have been treated (Rouzaud, 1994). At last, the Reynolds number of the flow based upon the plate length is equal to RESULTS The numerical predictions of the standard approach (Prandtl equations) and of the defect approach (system (9)) are compared to a reference solution. The three codes use the same turbulence model, i.e. the Baldwin-Lomax model. The comparisons are made for the longitudinal velocity profiles and the skin friction coefficient evolution along the wall. The velocity profiles are plotted at the dimensionless point x/l = 0.95, where L stands for the length of the plate. For the linear profile (fig. 3 and 5), the defect approach provides not only a good behaviour of the velocity at the edge of the boundary layer, but also a better estimate of the skin friction coefficient. The remarks are similar for the hyperbolic profile (fig. 4 and 6). standard boundary layer defect approach standard boundary layer defect approach Figure 3. Skin friction evolution Figure 4. Skin friction evolution

10 10 O. ROUZAUD ET AL. standard boundary layer defect approach Euler standard boundary layer defect approach Euler Figure 5. Velocity profiles Figure 6. Velocity profiles 4. Coupling methods In this section, improvements of the previous results are proposed either for the standard approach (Prandtl turbulent equations) or for the defect formulation (recombined system(9)). It could be possible to solve the thirdorder systems of the asymptotic model, but, for the sake of simplicity, we chose coupling methods. For each approach, a coupling method is proposed. Preliminary to this work, numerical experiments were performed in the laminar case for the two coupling methods and good results were obtained (Rouzaud, 1994) COUPLING FOR THE STANDARD APPROACH The method (fig. 7) is based on the work of Sawley and al. (1991). Once the first-order inviscid flow is given, the first-order boundary layer is computed from the Prandtl turbulent equations. At the boundary layer edge, the velocity is matched with the inviscid velocity U 1 (x,0). From this solution, a thickness δ.99 and a blowing velocity are defined. The thickness is the height at which the longitudinal velocity is equal to 99% of the inviscid velocity U 1 (x,0). The blowing velocity represents the displacement effect of the boundary layer on the inviscid flow. The perturbed inviscid flow is then computed by solving the Euler perturbation equations similar to system(2). At the wall, the inviscid normal velocity is equal to the blowing velocity. Afterwards, a new calculation of the boundary layer is made. The edge of

11 DEFECT FORMULATION 11 (U 1,V 1,P 1 ) First-order inviscid flow (U,V,P) Perturbated inviscid flow (U,V,P ) First-order inviscid flow (U,V,P) Perturbated inviscid flow u(x,y) -> U (x,0) 1 Blowing velocity u(x,y) -> U(x, δ) u(x,y) -> U (x,y) 1 Blowing velocity u(x,y) -> U(x,y) First-order boundary layer Coupled boundary layer First-order boundary layer Coupled boundary layer Prandtl equations Prandtl equations defect equations defect equations Figure 7. Coupling - Stand. approach Figure 8. Coupling - Defect approach the boundary layer is fixed at the height δ.99 and the edge conditions are the pressure P(x,δ) and the longitudinal velocity U(x,δ) of the perturbed inviscid flow (Sawley and al., 1991). Such a method is expected to give correct matchings for the pressure or the longitudinal velocity, but the continuity of the normal velocity or the slope of the longitudinal velocity are not ensured. However, if the corrections of the perturbed boundary layer are not too important, the slope u is correctly estimated. The results on the linear profile and on the hyperbolic profile are presented. The first-order solution is the standard boundary layer solution and is compared with the perturbed solution (). In both cases, the evolution of the skin friction coefficient (fig. 9, 10) is improved, the coupled results being close to the solution. For the velocity profiles (fig. 11, 12), the first-order inviscid solution (Euler 1) and the perturbated inviscid flow (Euler 2) are plotted. By comparison between the two cases, the effect of displacement due to the boundary layer is more important in the hyperbolic example than in the linear case. The coupled boundary layer result is still better than the first-order solution, in particular at the edge of the boundary layer COUPLING FOR THE DEFECT APPROACH For the defect formulation, the process is similar to the previous one (fig. 8) except that the boundary layer equations correspond to the recombined system (9) and that the boundary condition at the edge of the boundary layer is modified. Considering the skin friction coefficient behavior (fig. 13, 14) or the velocity profiles (fig. 15, 16), we notice that the first-order results are only slightly improved. This is due to the fact that the first-order solutions are already quite correct.

12 12 O. ROUZAUD ET AL. Figure 9. Skin friction coefficient Figure 10. Skin friction coefficient Euler 1 Euler 2 Euler 1 Euler 2 Figure 11. Velocity profiles Figure 12. Velocity profiles 5. Conclusion The defect formulation applied to the asymptotic analysis of Yajnik-Mellor gives a better description of the turbulent boundary layer than the standard approach does. On both theoretical and numerical grounds, the improvements are important and three main points are to be retained.

13 DEFECT FORMULATION 13 Figure 13. Skin friction evolution Figure 14. Skin friction evolution Euler 1 Euler 2 Euler 1 Euler 2 Figure 15. Velocity profiles Figure 16. Velocity profiles The first one is about the value of the pressure. Inside the boundary layer, the pressure may depend on the height y, although in the standard approach, the pressure is not dependent on the height y at the first two orders. The second point concerns the matching between the external flow and the outer layer. At first-order, the matching is already correct and the asymptotic model can handle non-linear variations of the inviscid flow. At last, the defect formulation allows the establishment of a more general

14 14 O. ROUZAUD ET AL. velocity law, taking into account the variations of the external flow with the y coordinate and the logarithmic behaviour is carried back to the defect velocity. Coupling methods are also proposed for the standard and defect formulations, the most interesting improvements being achieved for the standard approach. By the way, further numerical attempts are needed in order to find the range of validity of the two methods. In the future, the extension of the defect formulation to the turbulent boundary layer submitted to strong adverse pressure gradients or to the compressible turbulent boundary layer could be interesting. In fact, we believe the extension is possible as far as an asymptotic model based uponthe standard approach exists. However, due to the numerous asymptotic models proposed for each problem, the first step is to find the most appropriate asymptotic model. References Aupoix, B., Brazier, J-Ph. and Cousteix, J. (1992) Asymptotic defect boundary-layer theory applied to hypersonic flows, AIAA Journal 30, Brazier, J-Ph. (1990) Etude asymptotique des équations de couche limite en formulation déficitaire, PhD Thesis, ENSAE. Mellor, G. L. (1972) The large Reynolds number asymptotic theory of turbulent boundary layers, Int. J. Engineering Sciences 10, Rouzaud, O. (1994) Couches limites déficitaires en écoulement turbulent, PhD Thesis, ENSAE. Sawley, M. L. and Wuthrich, S. (1991) A coupled Euler/boundary layer method for nonequilibrium, chemically-reacting hypersonic flows, IMHEF, Report T Yajnik, K. S. (1970) Asymptotic theory of turbulent shear flows, J. Fluid Mechanics 42, part 2,

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