Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 41, 1, pp. 3-18, Warsaw 2003 MODELLING OF TURBULENT FLOW IN THE NEAR-WALL REGION USING PDF METHOD Marta Wacławczyk1 Jacek Pozorski Institute of Fluid-Flow Machinery, Polish Academy of Sciences, Gdańsk e-mail: mw@imp.gda.pl The paper presents near-wall turbulence models which incorporate the idea of elliptic relaxation. The simplified elliptic blending model is ap- plied in the Lagrangian probability density function (PDF) approach. The PDFmethod is extended to compute near-wall viscousmomentum transport. Computations are performed for fully developed turbulent channel flow and validated against available DNS data. Key words: near-wall turbulence, PDFmethod, elliptic relaxation 1. Introduction One of the inherent difficulties in modelling the turbulent flow is related to the near-wall regions. At the same time, most of the technically important turbulent flows are bounded, at least in part, by solid surfaces. In the im- mediate vicinity of the wall experimental investigations and the DNS results show the existence of complicated vortical structures of considerable kinetic energy (Aubry et al., 1988). DNS computations give insight into the dynamics of the turbulent eddies, mechanisms of their generation, and interactions be- tween them. However, due to high numerical cost of such simulations, engine- ering applications to date are limited to the Reynolds averaged Navier-Stokes (RANS)methodswhich provide statistical description of turbulent flows. The mean (ensemble averaged) variables are also affected by the presence of walls. In particular, the wall effects should be accounted for within RANS models. 1The author won the first prize awarded at the biennial young researchers’ con- test for the best work presented at the 15th Polish Conference on Fluid Mechanics, Augustów, September 2002 4 M.Wacławczyk, J.Pozorski First, themolecular transport of heat andmomentumbecomes important and cannot be neglected, as is sometimes done for high-Re turbulent flows far from solid boundaries. Due to the no-slip condition large gradients of mean stati- stics occur in the vicinity of the wall. Another effect is the lack of separation of macro- and microscales of turbulence which also arises from the viscosity action in the near-wall region. Moreover, the Reynolds stresses are strongly anisotropic which is caused by the blocking of wall-normal fluctuations. Most often, the modelling of near-wall flows is performed in the Eulerian approach, and so-called low-Re models are introduced for the purpose. They can be based on the damping functionmethod (Rousseau et al., 1997) or ellip- tic relaxation model of Durbin (1993). The functions which damp particular terms in the equations are derived from comparison with experiments, and often involve wall distance as an argument. For this reason, the damping func- tion approach is likely to fail in more complex geometries or complicated flow cases (e.g. with separation or reattachment zones). The elliptic relaxationme- thod is based on the Poisson equation for pressure fluctuations. The method accounts for the non-local character of pressure fluctuations and is therefore sounder from the physical point of view in comparison to the damping func- tions approach. The paper presents a model for near-wall turbulent flows derived for the Lagrangian (PDF) approach; there, the non-local wall effects should also be included.APDFmodel for low-Re numberswas derived byDreeben andPope (1998). In the model, viscosity was introduced through the Brownian motion in physical space and some additional terms in the equation for velocity. Non- local effectswere originallymodelled by the full six-equation elliptic relaxation method. However, in our work we apply a simplified approach of Manceau andHanjalić (2002), derived for theEulerianReynolds stress transportmodel (RSM). The method is adapted here to the Lagrangian PDF approach. Due to numerical problemswith down-to-the-wall integration, the previous scheme developed for high-Re turbulent flows (Minier and Pozorski, 1999) had to be changed and is now based on the exponential form of stochastic equations. The computations have been performed for the fully developed channel flow. The DNS data of Moser et al. (1999) are used for comparison. 2. Modelling of near-wall flows: elliptic relaxation method The instantaneous turbulent velocity field is influenced considerablyby the wall proximity. As a consequence, the wall effects have also an impact on the Modelling of turbulent flow... 5 flow statistics, like the mean velocity 〈Ui〉, the turbulent kinetic energy k, or the turbulent stresses 〈uiuj〉. Let us recall here that, according to the Reynolds decomposition, the instantaneous velocity can be written as a sum of itsmean and fluctuation parts Ui = 〈Ui〉+ui. Thewall boundary condition 〈Ui〉=0 leads to large velocity gradients and consequently to large values of the turbulence production term P = ∂〈Ui〉 ∂xj 〈uiuj〉 (2.1) (with summation over repeating indices) in the near-wall region. Due to the no-slip and impermeability conditions all components of velocity fluctuation are zero at the wall. Hence, in its proximity the components can bewritten as the Taylor series expansion (see eg. Manceau et al., 2001) u∼ a1y+a2y2+ ... v∼ b1y+ b2y2+ ... w∼ c1y+ c2y2+ ... where the streamwise, wall-normal and spanwise fluctuation components are denoted by u, v and w, respectively. Applying the above formulae to the continuity equation we obtain 0= ∂v ∂y ∣∣∣∣∣ y=0 ∼ b1+2b2y (2.2) hence b1 =0 and v∼ b2y2+ ... This result reveals the effect of kinematic blocking of the wall-normal fluctu- ations, which is felt even far from thewall and consequently introduces strong anisotropy to theReynolds stress tensor. On the other hand, the enhancement of the pressure fluctuations, due to their reflection from the surface, induces isotropisation of the Reynolds stresses. However, this effect is weaker than the kinematic blocking. As the wall is approached and the viscous transport becomes dominant, characteristic length and time scales of turbulent eddies become comparable with those of dissipative eddies. Hence, the Kolmogorov hypothesis is not valid in this region; this fact irrevocably limits the validity of standard turbulence models. In the case of incompressible flows considered here, the kinematic effects of wall blocking and pressure reflection are definitely of non-local elliptic nature 6 M.Wacławczyk, J.Pozorski and are immediately felt far from the wall. They represent a major challenge for turbulencemodels where only one-point closures (i.e. functions of only one point x of the flow) are involved. Let us recall here the transport equations for the Reynolds stresses (cf. Pope, 2000) which take the following form D〈uiuj〉 Dt =− ∂〈uiujuk〉 ∂xk︸ ︷︷ ︸ DT ij −〈uiuk〉 ∂〈Uj〉 ∂xk −〈ujuk〉 ∂〈Ui〉 ∂xk︸ ︷︷ ︸ Pij +Πij +ν∇2〈uiuj〉︸ ︷︷ ︸ Dν ij (2.3) where D/Dt stands for the material derivative along mean streamlines, the diffusion tensor DTij is connected with turbulent transport, D ν ij stands for the viscous transport and Pij is the production of turbulent stresses 〈uiuj〉. The non-locality of the flow field is represented by the tensor Πij which contains mean velocity-pressure gradient correlations and dissipation Πij =− 1 ̺ 〈 ui ∂p ∂xj 〉 − 1 ̺ 〈 uj ∂p ∂xi 〉 −2ν 〈∂ui ∂xk ∂uj ∂xk 〉 =φij + ǫij (2.4) The dissipation ǫij is a function of the fluctuating velocity gradient which can be interpreted as a quantity describing a ”short-range” non-locality, connec- ted with the length scales of dissipative eddies. Phenomena occurring at the smallest turbulent scales are difficult to model, especially when the hypothe- sis of separation of integral and viscous scales is no longer valid (Bradshaw, 1994). TheKolmogorov assumption breaks down in this region, as the rate of energy transfer from the large to the smaller eddies is not equal to the rate at which the energy is being dissipated by the smallest vortices. The pressure fluctuations p, present in the RHS of Eq. (2.4), can be computed from the elliptic Poisson equation (cf. Pope, 2000). Hence, they represent a long-range non-locality of turbulence. It is evident that non-local effects should be somehow included in turbu- lence models, whereas usual hypotheses applied to derive the basic version of closure are: high Reynolds number, local isotropy and quasi-homogeneity of turbulence. The tensor Πij is then a function of one-point statistics like the dissipation rate of the kinetic energy ǫ, turbulent stresses 〈uiuj〉 and mean velocity gradients. The dissipation ǫ is computed from its own transport equ- ation. In order to derive a physically sound closure for near-wall flows, Durbin (1993) proposed amodel which is based on an integral form of the tensor φij. From thePoisson equation for pressurefluctuations the tensor φij canbewrit- Modelling of turbulent flow... 7 ten as an integral containing a function of two-point statistics (i.e. non-local information) ̺φij(x)= ∫ Ω Ψij(x,x ′)GΩ(x,x ′) dV (x′) (2.5) where GΩ is a Green function of the flow domain Ω, further replaced by its free-space form G = −1/(4πr), with r = |x−x′|. The exact form of the function Ψij is detailed in e.g.Manceau (2000) andPope (2000). As evidenced by experiments, the two-point correlations can be approximated, for a wide range of r values, by exponential functions. Durbin proposed the following form for the function Ψij Ψij(x,x ′)= k(x) Ψij(x ′,x′) k(x′) exp ( − r L ) (2.6) where L is the characteristic length scale defined as the maximum of the turbulent length scale and the scale connected with dissipative eddies (valid close to the wall) L=CLmax {k3/2 ǫ ,CT (ν3 ǫ )1/4} (2.7) where CL and CT are model constants. Now, integral (2.5) becomes ̺ φij(x) k(x) =− ∫ Ω Ψij(x ′,x′) k(x′) exp(−r/L) 4πr dV (x′) (2.8) The term G′ = −exp(−r/L)/(4πr) which appears inside the integral, is the Green function connected with the operator 1/L2 −∇2; therefore φij is the solution of the following elliptic Helmholz equation L2∇2φij k − φij k =− φhij k (2.9) Above, φhij denotes a standard quasi-homogeneous model used to compute turbulent fields far fromwalls, e.g. Rotta’s return to isotropy or isotropisation of production (IP)model. It is assumed that far fromwalls the Laplacian term in Eq. (2.9) disappears and then φij is equal to its quasi-homogeneous form. The same elliptic equation is solved also for the dissipation tensor ǫij and hence for the tensor Πij. A simplified elliptic relaxation approach was specified by Manceau and Hanjalić (2002). They state that six elliptic equations (2.9) of the originalmo- del of Durbin are somewhat redundant and unnecessarily increase the compu- 8 M.Wacławczyk, J.Pozorski tational cost.Manceau andHanjalić solve only one additional elliptic equation for the so-called blending function α L2∇2α−α=− 1 k (2.10) The velocity-pressure-gradient tensor φij is then found from an interpolation between its near-wall and quasi-homogeneous limits φij =(1−kα)φwij +kαφhij (2.11) It follows from the above formula that the required near-wall value of kα is 0; in the core region of the flowwe expect kα=1. Hence, Eq. (2.10) is solved with the following boundary conditions α ∣∣∣ y=0 =0 α ∣∣∣ y=H = 1 k (2.12) The same blendingmethod is used for the dissipation tensor ǫij. 3. Turbulence modelling using PDF method The modelling of a turbulent field can be performed in two basic appro- aches, i.e. in the Eulerian or Lagrangian point of view. In the first one, most often used, flow variables are connected with a certain point in space (x,y,z) and time t. Thus, discretized equations can be solved on a space-time grid. A good example of the Eulerian approach are the Reynolds stress equations (2.3), presented in the previous section. In the Lagrangian approach flow pa- rameters are related to a certain element of fluidwhich has the initial position (x0,y0,z0) in the time instant t0. In the Lagrangian approach used in the paper we solve equations for stochastic particles which model fluid elements. 3.1. High-Reynolds models In themodelling of high-Reynolds number flows, viscosity is not accounted for explicitly, the viscous action is modelled only by the dissipation ǫ. Trans- port equations for stochastic particles take the general form (Pope, 2000) dXi =Uidt (3.1) dUi =− 1 ̺ ∂〈P〉 ∂xi dt−Gij(Uj −〈Uj〉)dt− 1 2 ǫ k (Ui−〈Ui〉)dt+ √ BdWi Modelling of turbulent flow... 9 where B= 2 3 Gkl〈ukul〉 Above, dW is an increment of the Wiener process, written in the discrete form as ∆W = √ ∆tξ where ξ is a standard random Gaussian number. The turbulence model is introduced by a specific form of the tensor Gij which is a function of mean turbulence statistics and model constants. In the Monte Carlo simulation stochastic differential equations (3.1) are solved for a large set of stochastic particles. In order to compute themean statistics theflowdomain is first discretized and then parameters connected with particles within one cell of the spatial grid are averaged. PDF computations in the high Reynolds number approach for fully developed channel flow were performed by Minier andPozorski (1999). Boundary conditions for stochastic particles were placed in the logarithmic region. It is important to note that from Lagrangian equations (3.1) one can de- duce correspondingEulerian equations for the one-point statistics, namely the mean velocity 〈Ui〉, the Reynolds stresses 〈uiuj〉, as well as for higher order moments, e.g. triple correlations 〈uiujuk〉. For this purpose we write the evo- lution equation for the probability density function f = f(V ;x, t), also called the Fokker-Planck formula (van Kampen, 1990). For the sake of an example, let us consider the following stochastic equations dX =Udt dU =Adt+BdW where Aand B are constants.Theprobabilitydensity function connectedwith the above equations is denoted by f(V ;x,t), where x, V belong to a sample space of the position X and the velocity U. The expression f(V ;x,t)dV is the probability that the variable U connected with a stochastic particle takes a value within the bounds V ¬U ¬ V +dV . The evolution equation for the probability density function writes ∂f ∂t +V ∂f ∂x =−A ∂f ∂V + 1 2 B ∂2f ∂V 2 Similarly, stochastic equations (3.1) correspond to the following formula for the PDF ∂f ∂t +Vi ∂f ∂xi = 1 ̺ ∂〈P〉 ∂xi ∂f ∂Vi + ∂ ∂Vi [ Gij(Vj −〈Uj〉)f ] + (3.2) + 1 2 ǫ k ∂ ∂Vi [(Vi−〈Ui〉)f]+ 1 2 B ∂2f ∂V 2j 10 M.Wacławczyk, J.Pozorski Themean velocity and other statistics can be computed from integration over the sample space 〈Ui〉(x, t)= +∞∫ −∞ Vif(V ;x, t) dV (3.3) 〈uiuj〉(x, t)= +∞∫ −∞ (Vi−〈Ui〉)(Vj −〈Uj〉)f(V ;x, t) dV In order to derive the transport equation for the mean velocity, formula (3.2) is multiplied by Vi and then integrated over V . Equations for the Reynolds stresses are obtained after multiplying (3.2) by (Vi − 〈Ui〉)(Vj − 〈Uj〉) and integrating. As a result we get D〈Ui〉 Dt =−∂〈uiuj〉 ∂xj − 1 ̺ ∂〈P〉 ∂xi (3.4) D〈uiuj〉 Dt =− ∂〈uiujuk〉 ∂xk −〈uiuk〉 ∂〈Uj〉 ∂xk −〈ujuk〉 ∂〈Ui〉 ∂xk − −Gjk〈uiuk〉−Gik〈ujuk〉− ǫ k 〈uiuj〉+Bδij At this stage, the PDF method corresponds to the Eulerian high-Re mean velocity equation and the Reynolds stress transport (RSM) models; however the turbulent transport term ∂〈uiujuk〉/∂xk is exact and does not require modelling. The particular form of Gij depends on an assumed turbulence model (Pope, 1994). As an example, Gij = ǫ/2k(1−2C̃)δij where C̃ is a constant, corresponds to the Eulerian return-to-isotropy model derived by Rotta. 3.2. Low-Reynolds models In the modelling of low-Reynolds numbers flows it is important to inc- lude viscous transport of momentum. This poses no particular difficulty in the Eulerian approach, where viscosity appears explicitly in terms containing Laplacians of mean quantities. A low-Re model for the Lagrangian approach was derived by Dreeben and Pope (1998); first, the viscous diffusion term was represented through a random motion of stochastic particles. Hence, the equation for a particle position writes dXi =Uidt+ √ 2νdWXi (3.5) Modelling of turbulent flow... 11 This form is chosen to retrieve the ν∇2f term in the evolution equation for the PDF; this further leads to the Laplacian terms in themean velocity and stress transport equations. Next, the expression for velocity increment is derived via the Ito equation (van Kampen, 1990) dUi = ∂Ui ∂t dt+ ∂Ui ∂xj dXj + 1 2 ∂2Ui ∂x2k (dXk) 2 (3.6) After substituting formula (3.5) the above equation takes the form dUi = (∂Ui ∂t +Uj ∂Ui ∂xj ) dt+ √ 2ν ∂Ui ∂xj dWXj +ν ∂2Ui ∂x2k dt (3.7) Noting that the expression in parentheses is the RHS of Navier-Stokes equ- ations, we get dUi = ( − 1 ̺ ∂P ∂xi +ν ∂2Ui ∂x2k ) dt+ √ 2ν ∂Ui ∂xj dWXj +ν ∂2Ui ∂x2k dt (3.8) The instantaneous velocity and pressure can bewritten according to the Rey- nolds decomposition as a sum of their mean and fluctuation parts. However, gradients of ui and p are unknown and require modelling. Hence, for these terms we apply the same closure as for high Reynolds numbers dUi = ( − 1 ̺ ∂〈P〉 ∂xi +2ν ∂2〈Ui〉 ∂x2k ) dt+ √ 2ν ∂〈Ui〉 ∂xj dWXj + (3.9) +Gij(Uj −〈Uj〉)dt− 1 2 ǫ k (Ui−〈Ui〉)dt+ √ BdWVi Next, a formula for the PDF corresponding to equations (3.5) and (3.9) can be derived. After the proper integration we obtain the following transport equations D〈Ui〉 Dt =− ∂〈uiuj〉 ∂xj − 1 ̺ ∂〈P〉 ∂xi +ν ∂2〈Ui〉 ∂x2k (3.10) D〈uiuj〉 Dt =− ∂〈uiujuk〉 ∂xk −〈uiuk〉 ∂〈Uj〉 ∂xk −〈ujuk〉 ∂〈Ui〉 ∂xk + +ν ∂2〈uiuj〉 ∂x2k −Gjk〈uiuk〉−Gik〈ujuk〉− ǫ k 〈uiuj〉+Bδij It shouldbenoted thatReynolds equation (3.10)1 is exact andall the transport equations contain the required Laplacians of mean quantities. 12 M.Wacławczyk, J.Pozorski 3.3. Modelling of near-wall flows Apart from the viscosity action, the modelling of near-wall flows should also account for non-local kinematic effects whichwere described in Section 2. For the purpose,Dreeben andPope (1998) solved elliptic relaxation equations for all components of the tensor Gij. In our work we applied the simplified version of themethod (Manceau andHanjalić, 2002)where only one additional equation for elliptic blending function (2.10) is solved. Components of the tensor Gij are then computed from the relation Gij =(1−kα)Gwij +kαGhij (3.11) The elliptic blendingmethod was initially derived for the Eulerian approach. Here, we apply it to the Lagrangian equations. One of the differences is con- cerned with the proper near-wall form Gwij. It should assure a proper scaling of Reynolds stresses in the near-wall region, namely 〈u2〉 ∼ y2, 〈v2〉 ∼ y4, 〈w2〉 ∼ y2 and 〈uv〉 ∼ y3. Here, we recall a derivation proposed by Dreeben and Pope (1998). Near the wall, the turbulent transport, convective and pro- duction terms become negligible andReynolds-stress equations (3.10)2 reduce to ν ∂2〈uiuj〉 ∂y2 − 〈uiuj〉 k ǫ=Gik〈ujuk〉+Gjk〈uiuk〉− 2 3 Gkl〈ukul〉δij (3.12) It is assumed that near the wall the tensor Gij takes the form Gij = Cǫ/k where C is a constant. From the near-wall balance of terms in the kinetic energy equation ν∇2k = ǫ we derive the scaling formula ǫ/k = 2ν/y2 which is further used in (3.12), leading to the following differential equation ν ∂2〈uiuj〉 ∂y2 −aij 〈uiuj〉 y2 =O(y) (3.13) (no summation over repeating indices). The solution to the above equation is 〈uiuj〉=Aijy(1− √ 1+4aij)/2+Bijy (1+ √ 1+4aij)/2+Cijy 3 (3.14) The value of exponent in the first term is negative, hence the no-slip boundary condition forces Aij = 0. In order to assure that 〈u2〉 ∼ y2 and 〈w2〉 ∼ y2 we should have a11 = a33 = 2; with aij > 6 the last term on the RHS will dominate the solution and 〈uiuj〉 ∼ y3. The boundary form Gwij used in the present computations Gw22 = 9 2 ǫ k Gwij =0 for i 6=2 or j 6=2 (3.15) Modelling of turbulent flow... 13 gives a11 = a33 = 2, a12 = 11, a22 = 14, which assure the proper scaling of all Reynolds stresses except for 〈v2〉, which should be of the order y4. It is a drawback of the elliptic relaxationmethod that it does not provide the proper scaling of all Reynolds stresses. We also state that with the above definition, Gwij does not contract to zero when the equation for the kinetic energy is derived, however the remaining term is of a smaller order and can therefore be neglected. 4. Numerical results The computations have been performed for the case of fully developed turbulent channel flow at Reτ = 395. In the presence of large mean velocity gradients it was necessary to use a turbulence model that accounts for the rapid pressure term (Pope, 2000). For this reason, we implemented a model that corresponds to theEulerian basic pressure-strainmodel (Rotta+IP) used by Durbin (1993), with the same values of constants. The tensor Ghij is Ghij = 1 2 ǫ k (C1−1)δij −C2 ∂〈Ui〉 ∂xj (4.1) with C1 = 1.5 and C2 = 0.6. For this case, we do not solve a separate equation for the turbulent frequency ω, but the values of the turbulent time scale T =1/〈ω〉 are read froma filewith theDNSdata ofMoser et al. (1999). It is left for further work to set the proper values of coefficients in the ω equation and solve it together with the IPmodel for velocity. The dissipation which appears in the model equations is a sum of two components ǫ= 〈ω〉k+C2Tν〈ω〉2 (4.2) the first of them stands for the turbulent time scale and the second one is connected with the scale of dissipative eddies. The results illustrate the need of a specificnear-wall treatment, as actually done through the elliptic blending equation. Both the mean velocity (Fig.1a) and the turbulent kinetic energy (Fig.1b) are in better accordance with the DNS data when the elliptic rela- xation model is applied. In Fig.1b, a sharpmaximum of the kinetic energy is observed at y/H = 0.03 (around y+ ≈ 12) as evidenced by the DNS data, although it is still somewhat underpredicted. Contrary to the streamwise component u, the fluctuations of v are dam- ped. This is illustrated by scatter plots of near-wall streamwise, spanwise and 14 M.Wacławczyk, J.Pozorski Fig. 1. Turbulent channel flow at Re τ =395, the IPmodel: (a) mean velocity 〈U〉+, (b) turbulent kinetic energy k+. DNS data (Moser et al., 1999) ■; PDF computations: without elliptic relaxation (– – –), with elliptic relaxation (——–) Fig. 2. Turbulent channel flow at Re τ =395; Scatter plots of velocity components near the wall: (a) streamwise (solid line – theoretical profile U+ = y+), (b) wall-normal, (c) spanwise Modelling of turbulent flow... 15 wall-normal velocity components presented in Fig.2. The variance of wall- normal fluctuations is much smaller than that of the two other components. After applying the elliptic relaxation method also the shear stresses 〈uv〉 are in a good overall agreement with the DNS data (cf. Fig.3a). This is also clearly seen in the near-wall scaling presented in Fig.3b. Although the elliptic relaxation improves the results, we do not obtain exactly 〈uv〉 ∼ y3. This can be caused by numerical problems with the near-wall integration. Fig. 3. Turbulent channel flow at Re τ =395; (a) turbulent shear stress 〈uv〉, (b) near-wall scaling of turbulent shear stress and kinetic energy. DNS data (Moser et al., 1999): symbols; PDF computations: without elliptic relaxation (– – –), with elliptic relaxation (——–) Let us only note here that in the vicinity of the wall, the turbulent kinetic energy k tends to 0 as y2, while its dissipation rate attains a constant value. At the same time the streamwise and spanwise velocity components scale as y. As a consequence, when the Euler discrete scheme is used to solve stochastic differential equation (3.9), two of its components tend to infinity with y→ 0 ǫ k ui∆t∼ 1 y →∞ for i 6=2 (4.3) This causes serious numerical problems in the near-wall integration, unless the time step ∆t is very small.This is the reason for introducinganothernumerical scheme based on the exponential solution to equations (3.9). The numerical schemepresented in theworkofMinier et al. (2001) hasbeen furtherdeveloped here to account for a non-diagonal form of thematrix Gij and included in the numerical algorithm. 16 M.Wacławczyk, J.Pozorski 5. Conclusions and perspectives The elliptic relaxation method was used to model the non-local effects connected with the presence of the wall. The derivation of the method was briefly recalled in the paper. An advantage of the elliptic relaxation in compa- rison to the damping function approach is that there is no explicit dependence on the wall distance y or local Reynolds number. Moreover, the method is more physically sound and does not depend on flow geometry. Simplified va- riants of themethod like e.g. k-v2-f (Durbin, 1995) or elliptic blendingmodel (Manceau andHanjalić, 2002) are interesting for engineering applications due to a reduced numerical cost. Another promising perspective for the near-wall RANS models is to solve the equations in conjunction with the Large Eddy Simulation (LES) approach for the outer layer of the flow (Piomelli andBala- ras, 2001). This makes it possible to perform high-Re LES computations at a reasonable cost. In the paper we applied the elliptic blending approach in the Lagrangian PDFmodel to compute velocity statistics in a fully developed turbulent chan- nel flow. Reasonable accuracy has been achieved in comparison with the ava- ilableDNSdata ofMoser et al. (1999) at Reτ =395.Thenear-wall turbulence modelling in the PDF approach is, to a certain degree, related to the Eulerian second-moment closure. However, the turbulent transport term is closed and does not requiremodelling. ThePDFmethod is often used tomodel chemical reactions (Libby and Williams, 1994) due to the closed source term. For the purpose, either a joint velocity-scalar PDF approach can be used or velocity statistics can be taken fromexternal datawith only scalar dynamics computed by the PDFmethod (Pozorski, 2002). When supplemented with a suitable scalar transport equation, the appro- ach presented in the paper can be applied to the case of near-wall turbulence with heat transfer to model the thermal fluctuations in the vicinity of the wall. 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Zaprezentowano przy tym model dla funkcji gęstości prawdopo- dobieństwa (ang. PDF — Probability Density Function) stosowany do przepływów o niskich liczbach Reynoldsa.Wykonano obliczenia dla przypadku przepływu turbu- lentnego w kanale płaskim; wyniki porównano z dostępnymi danymi DNS. Manuscript received October 22, 2002; accepted for print November 22, 2002