Internal Report 00/4 On the Wall Boundary Condition for Turbulence Models JONAS BREDBERG Department of Thermo and Fluid Dynamics HALMERS NIVERSITY OF ECHNOLOGY C ¢¡ U T £¥¤§¦©¨ £ ¦©¦¥ ¥ On the Wall Boundary Condition for Turbulence Models JONAS BREDBERG Department of Thermo and Fluid Dynamics Chalmers University of Technology ABSTRACT This report explains and discuss two main boundary condi- tions for turbulence models: the integrating and the wall fun- ction approach. The report thoroughly derives the law-of-the- wall for both momentum and thermal field. The deviation of these laws from DNS-data is discussed. The Wilcox ¢¡¤£ tur- bulence model constants are derived on the basis of the law- of-the-wall and experiments. Using a simplistic treatment, the low-Reynolds number modifications (damping functions) to the ¥¡¤£ model are explained. Included in the report is also a guideline on the implemen- tation of wall functions, both the standard and the Chieng- Launder two-layer variant. A new boundary condition for turbulence models combining wall function and integration approach is presented. Keywords: boundary condition, wall function, turbulence model ii Contents Nomenclature iv 1 Introduction 1 2 Near-wall Physics 1 2.1 Buffer Layer . 1 2.2 Viscous sub-layer . 1 2.3 Inertial Sub-layer . 2 3 Law-of-the-Wall 3 3.1 Momentum . 3 3.2 Energy/Temperature . 4 3.3 Turbulence . 6 3.4 Turbulence Model Constants . 7 3.4.1 Coefficient, ¢¡ . 7 3.4.2 Coefficient, . 7 3.4.3 Coefficients, £¥¤¦£§¡ . 7 3.4.4 Coefficient, ¨ . 7 3.5 HRN Turbulence Model . 8 4 Modelling Near-wall Turbulence 8 4.1 LRN-models: Integration Method . 8 4.1.1 The damping functions . 9 ¢¡ 4.1.2 © modification, . 9 4.1.3 Production-to-dissipation rate . 10 ¡ 4.1.4 Rationale behind the and coefficients 10 4.2 HRN-models: Wall Functions . 11 4.2.1 Standard wall functions . 11 4.2.2 Launder-Spalding methodology . 11 4.2.3 Improvements of the near-wall represen- tation, Chieng-Launder model . 12 4.2.4 Modification to the Chieng-Launder model 13 4.2.5 Extensions to the Chieng-Launder model 14 4.3 New Model: Blending Integrating and Wall Function Boundary Conditions . 14 4.3.1 Blending function . 15 4.3.2 HRN part: Simplified Chieng-Launder ¡¤£ model applied to a -type turbulence model . 15 4.3.3 LRN part: Standard ¥¡¤£ . 16 4.3.4 The blending ¥¡¤£ turbulence model . 16 5 Modelling Near-wall Heat Transfer 18 5.1 Integration Method . 18 5.2 Wall Function Method . 19 5.3 New Blending Model . 19 iii Nomenclature Greek Symbols Latin Symbols ¡ +,¨-. /¢ Thermal diffusivity ¡£¢ Turbulence model constant ¡ ¤ ¡ ¡ ¡£¢ Turbulence model coefficients ¥¤ ¡§¦©¨ ¢ © Specific heat ¤ ¡ ¡£¢ Turbulence model coefficients ¡ ¡ ¡£¢ © Various constants ¡ ¡£¢ ¨ Turbulence model coefficient k ¡ ¡£¢ © Length-scale constant "¢ Characteristic length scale ¡ l © ¡£¢ Turbulence model coefficient ¡ "¢ Cell size ¡ £ ¥¨ ¡ ¡£¢ © Skin friction coefficient, ¡ ¨4=m¢ R Dissipation rate ¡ ¢ Hydrualic diameter n ¡ ¡£¢ ! Normalized wall distance ¡ "¢ Rib-size o # ¡£¢ Van Karman constant ¡ p 9E ¡£¢ Turbulence model constant ¡ $ ¤q ¨ ¡ +,¨G ¢ Thermal conductivity, © ¡£¢ Damping function ¡ $ ¡ ¨4.¢ M Kinematic viscosity 7 % ¤'& ¤)( ¡ ¡£¢ ¤ Functions * q 4-¨:¢ Dynamic viscosity ¡ "¢ Channel half-height ¡ * = ¡ c ¨- ¢ Density +,¨-. /¢ Heat transfer coefficient ¡ 0 £ 4 Z\[:¢ Specific dissipation rate ¡ "¢ Channel height ¡ 1 £ ¡ ¡ ¡£¢ £¥¤ Turbulence model coefficients 32 ¡ ¡£¢ ¤ Tensor indices, ¡ 4m¢ Time streamwise: 1 (U) 7 ¡ ¨-G:¢ Shear stress 8 7 wall normal: 2 (V) X ¡ ¨G:¢ Wall shear, spanwise: 3 (W) ¡ ¨45¢ Turbulent kinetic energy 6 Superscripts ¡ ¢ Length scale ! 78 Effective value r?s 8 ¡ ¡£¢ Nusselt number X 7 9 Normalized value using : s t _ _ _ ¡ ¨:¢ Static pressure X 9 ¨ s t 8 ¡ "¢ Rib-pitch X 9<; u¨ s t 8 ¡ ¨4=5¢ Turbulent production, X @ U U ¨M 8?>@A8?> B s t 8 B.C C?D ¨ X O¨ 9E s t 8©v ¡ ¡£¢ Prandtl number X R RWMO¨ 9E s t 8 ¡ ¡£¢ Functional Prandtl number X @ £ £MO¨ F s t 8 ¡ +,¨-G5¢ Heat flux X ©¨ HJI H; HLK s t 8?w 8 w 8 > > ¤ ¤ ¡ ¡£¢ Turbulence model constants X ¨ H 0 ! r ¨M ¡ ¡£¢ Reynolds number, V H ¡ !N Normalized value using : v ¡ ¡£¢ t Turbulent Reynolds number, V [)x N U ¡ U © O¨MyP ¡zRQ O¨P £MQ ¨ PSRMOQ M ¨M v , , t V H !T ¡ ¡:[)x U U ¨MyP ¡¤£Q ¡£¢ Viscous sub-layer Reynolds ¡ TV T ¨WM number, U H 8 !X X ¡£¢ ¡ Subscripts based Reynolds f 8 X M U Non-modified value number, ] @ Y B 4 Z\['¢ ¡ First interior node Strain-rate tensor, { @ @ ] B^ B C C?D C C?D ¨ ¨ Q ¨OP Bulk value _ ©| ¢ ¡ Centerline value Temperature 0/H7 > ` ¢ ¡ HRN (logarithmic) part Fluctuating temperature 6 _ X /¢ ¡ Laminar quantity Friction temperature, H7 _ _ 8 X F | ¡ Qacb ¨ P LRN (viscous) part @ } ¡ /¨4.¢ Velocity North face value ~ @ > 8 /¨4.¢ ¡ Node value Fluctuating velocity ` @ 8d> 8d> B Turbulent quantity ¡ G¨45¢ Reynolds stresses _ 8 X Thermal quantity w .¥¨ ¡ ¨W4.¢ Friction velocity, e 8 8 ]gfWfhfji X X Viscous sub-layer value ¡ ¨W ¡ ¡£¢ Normalized value: Wall value D "¢ Streamwise coordinate ¡ Quantity based on the ¡ "¢ U Wall normal coordinate friction velocity iv 1 Introduction 2 Near-wall Physics One of the most common engineering problems is com- The near-wall region may be sub-divided into three dif- puting turbulent flows that are influenced by an adja- ferent areas, Tennekes and Lumley [31]: cent wall. Examples of this are flows in turbomachinery, s f¢¡ ¡¤£ around vehicles, and in pipes. The main two effects of a viscous sub-layer U wall are: s £¥¡ ¡§¦hf U buffer layer Damping the wall normal components, making the s ¦hf¥¡ ¡ fWf turbulent flow anisotropic. inertial sub-layer U Increasing the production of turbulence through the shearing mechanism in the flow. The turbulence is negligible in the viscous sub-layer, while the viscous effects are small in the inertial sub- The wall gives rise to a boundary layer, where the ve- layer. In the buffer layer, however, both turbulent and locity changes from the no-slip condition at the wall to viscous effects are of importance, see Sahay and Sreeni- its free stream value. The variation is usually largest in vasan [29] and Fig. 1. the near-wall region, and hence the strongest gradients are found here. Similarly, for heat transfer applications, 1.5 there exists a thermal boundary layer with equally large Total shear gradients. Because both heat transfer and friction are Turbulent shear computed using gradients of the dependent variable, it Laminar shear is very important to accurately capture this near-wall variation. 1 The standard method is to apply a very fine mesh close to the wall, to resolve the flow. This method is called the integration method, which necessitates an LRN (low- ¨ © Reynolds-number) type of turbulence model. At higher Reynolds numbers, the region under the wall influence 0.5 diminishes. However, because it is equallyPSfragimportantreplacementsto accurately capture the near-wall gradients, a large num- ber of nodes are then necessary. From an engineering point of view, this becomes inconceivable and a function that bridge the near-wall region is instead introduced, 0 thereby significantly reducing the computational requi- 0 10 20 30 40 50 rements. The anticipation is that this can be done with © only a small deterioration in the results. This latter ap- Figure 1: Laminar and turbulent shear in the near wall t H ¦ £ proach is denoted the wall function method, which uses region. Channel flow, DNS-data, !X [25]. an HRN (high-Reynolds-number) type of turbulence mo- del. 2.1 Buffer Layer The maximum turbulent production occurs in the buffer s t ] layer at roughly U , slightly dependent on the Rey- nolds number. Due to large variations in the different turbulence source terms, see Figs. 2 and 3, the model- ling becomes very difficult. Today there exists no general method for applying a turbulence model, with the first computational interior node located in the buffer layer. Instead of trying to model the behaviour in the buffer layer, the common practise is to place the first near-wall node in either the viscous sub-layer (LRN-models), or in the inertial sub-layer (HRN-models). A viscous appro- ach is valid in the former case, while a turbulent ap- proach is more correct in the latter, for the first interior computational cell. 2.2 Viscous sub-layer In the viscous sub-layer the following asymptotic re- lation for the velocity, temperature, turbulence kinetic 1 0.25 10 , DNS 0.2 9 Production , Model Dissipation 0.15 , DNS 8 , Model 0.1 , DNS 7 , Model ¡ 0.05 , DNS 6 ¥ PSfrag replacements , Model ¦ ¤ ¡£¢ 0 5 −0.05 4 −0.1 3 PSfrag replacements −0.15 2 −0.2 1 −0.25 0 0 10 20 30 40 50 0 2 4 6 8 10 © © _ ¤ ¤ ¥¤ R Figure 2: Production and dissipation in the -equation. t H ¦ £ Figure 4: Near-wall variation in . Channel flow, ! X t H ¦ £ Channel flow, DNS-data, [25]. DNS-data, !X [25]. 0.25 20 0.2 Viscous diffusion 18 Turbulent transport Pressure-diffusion 16 0.15 Sum of diffusion terms 14 0.1 12 ¦¨§ © 0.05 r 10 0 8 PSfrag replacements 6 −0.05 4 −0.1 2 PSfrag replacements 0 0 10 20 30 40 50 10 20 30 40 50 60 70 80 90 100 © s U Figure 3: Viscous diffusion, turbulent transport and _ ¤ ¤ ¥¤ R Figure 5: Variation in in the inertial sub-layer. t H ¦ £ ! pressure diffusion terms in the -equation. Channel X t H ¦ £ !-X Channel flow, DNS-data, [25]. Labels as in flow, DNS-data, [25]. Fig. 4 energy and dissipation rate is valid: the dissipation rate, while the turbulent kinetic energy s s t U (1) should be solved and not set a priori.