Computation of turbulent natural convection at vertical walls using new wall functions

Transcription

1 Computation of turbulent natural convection at vertical alls using ne all functions M. Hölling, H. Herig Institute of Thermo-Fluid Dynamics Hamburg University of Technology Denickestraße 17, 2173 Hamburg, Germany 1. Introduction Natural convection can be observed in technical applications, e.g. climatisation of buildings or cooling of electronic devices, as ell as in nature, e.g. oceanic or atmospheric flos. In this study, turbulent natural convection at heated or cooled alls is analysed asymptocically. Often, natural convection is studied at simple geometries as shon in Figure 1. A 1-D situation is encountered in an infinite vertical channel, see Versteegh and Nieustadt [1] as ell as Betts and Bokhari [2], for example. Natural convection along a vertical plate as investigated by Tsuji and Nagano [3, 4] among others. Ampofo and Karayanis [5] conducted experiments for a lo aspect ratio cavity ith side alls at different temperatures. All cases can be characterised by a Grashof number based on the channel idth h or the distance along the plate, L Gr = g xβ Th 3 ν 2 or Gr L = g xβ TL 3 ν 2. (1) In the near all region, the governing equations for turbulent natural convection reduce to = ( a T ) y y v T (2) = ( ν u ) y y u v + g x β (T T ) (3) see for example Tsuji and Nagano [3]. A classical asymptotic theory as published by George and Capp [6] in hich poer las for the temperature and velocity profiles are proposed. It is idely used, although Versteegh T hot g x x y vertical channel h T T u T hot > T cold T cold T vertical plate T > T T u T T hot cavity T adiabatic u T adiabatic h T cold Figure 1: Standard geometries for turbulent natural convection. Left: infinite vertical channel [1, 2]; Middle: vertical plate [3, 4]; Right: cavity ith differentially heated sidealls [5] 1

2 viscous sublayer outer layer (y η) all overlap layer inner layer (y y ) y Figure 2: To-layer-structure of the temperature field for turbulent natural convection and Nieustadt [1] and Henkes and Hoogendorn [7] shoed deviations beteen this theory and their data. An alternative approach as proposed by Yuan et al. [8], ith profiles obtained by dimensional considerations and curve fitting. Despite the intensive research in natural convection for the last decades there is no theory available today that can accurately describe the near all region of natural convection. Therefore, e ant to present ne results, first shon in Hölling and Herig [9] together ith their implementation as all functions in a CFD code. Results shon afterards are valid for fluids ith Pr = Temperature Profiles An expection of the energy equation, (2), shos that the heat flux consists of a molecular part (a T/ y) and a turbulent part ( v T ) ith its sum equal to the all heat flux (a T/ y ). Thus, the all heat flux can be identified as the crucial parameter in turbulent natural convection and a characteristic temperature T c should be based on it. Folloing dimensional considerations e introduce T c ( a 2 gβ T y 3 ) 1/4 and Θ T T T c (4) as the non-dimensional temperature Θ. Next, the all distance has to be non-dimensionalised. Analysing the temperature field, it turns out that it has a to-layer-structure as shon in Figure 2. Close to the all an inner layer exists in hich molecular and turbulent heat transfer are present. The region further aay from the all is identified as the outer layer hich is dominated by turbulent heat transfer. For Gr the ratio of these to layers tends to zero leading to a singular temperature profile at the all. Thus, to different non-dimensional all distances have to be introduced, one for the inner and one for the outer layer. The thickness δ of the inner layer scales as δ = T c T y and the non-dimensional all distance for the inner layer is y y/δ. For the outer layer a geometric length (channel idth h or distance along the plate L) can be used as a length scale, resulting in η y/h (or η y/l). As shon in Figure 2 the to layers are not sharply separated but an overlap layer exists in-beteen. In this overlap layer, the temperature field can be simultaneously described in terms of the inner layer (using y ) and the outer layer (using η). An asymptotic matching of gradients in this layer results in lim y y Θ y 2 1 (5) = lim η η Θ η. (6)

3 Θ Θ = C ln(y ) + D Θ = y y Figure 3: Non-dimensional temperature profiles of Betts and Bokhari [2] for Gr = ( ), from Ampofo and Karayiannis [5] for Gr = ( ), of Tsuji and Nagano [3] for Gr L = ( ), for Gr L = ( ), for Gr L = ( ), for Gr L = ( ), for Gr L = ( ), for Gr L = (+) and of Cheeseright [1] for Gr L = ( ). Profiles are shifted by one unit each for better clarity. This equation can only be fulfilled if both sides are equal to a constant C. Integration of equation (6) therefore leads to Θ = C ln ( y ) + D. (7) The constants C =.427 and D = 1.93 ere determined by comparison ith experimental and numerical data. Figure 3 shos temperature profiles from various studies in their nondimensional form. The agreement beteen the data and the logarithmic profile according to equation (7) is quite good. Additionally the temperature profile for the viscous sublayer is plotted. In this region adjacent to the all heat transport is purely molecular (see Figure 2) and the temperature profiles are Θ = y, folloing from equation (2). This again is in good agreement ith the data. Here, as ell as for the velocity profiles in the next section, variable property effects had to be taken into account hen constants ere determined from experimental data, see Hölling and Herig [9] for details. 3. Velocity Profiles In a straightforard approach, as applied by George and Capp [6], velocity gradients are matched in the overlap layer. This, hoever, results in a poor representation of the flo field [1, 7]. Instead, in our study the velocity profile in the overlap layer is directly derived from the momentum equation (3). With an eddy-viscosity approach, u v = ν t u/ y, together ith ν t ν equation (3) in non-dimensional form reads = ( νt y ν U ) y Θ + Θ (8) 3

4 ith U = u/u c as non-dimensional velocity based on u c = g xβtc 3 T ν y 2. (9) U can be obtained by integrating equation (8) ith all terms reritten as functions of y. Since the temperature profile is already knon in terms of y, see (7), only ν t /ν = f(y ) has to be found. First, this expression is reritten ith the turbulent Prandtl number σ t as ν t ν = σ t Pr a t a. (1) From the DNS data of Versteegh and Nieustadt [1] it is found that σ t is nearly constant in the overlap layer ith σ t =.9, like for forced convection. The ratio of the turbulent thermal diffusivity a t and the molecular thermal diffusivity a can be obtained by differentiating the temperature profile (7) and comparing it to the non-dimensional form of equation (2). From this a t a = y ν t C ν = σ t y Pr C. (11) No, all terms in equation (8) depend on y only and it can be integrated, from hich ( C [ ln(y ) 2 ] + D Θ U = CPr σ t y ) + E ln(y ) + F (12) results. The constants of integration, E and F, are constant ith respect to y but they depend on the non-dimensional all shear stress U / y : E =.49 U y 2.27 and F = 1.28 U y (13) This as found by comparison ith numerical and experimental data. Figure 4 shos non-dimensional velocity profiles from various studies. The velocity profiles in the overlap layer, equation (12), are a good representation (better than the profiles from George and Capp [6]) of the experimental data as long as the Grashof number is high enough. Additionally, the velocity profiles in the viscous sublayer are shon, obtained by integrating equation (3) ith u v =. In its non-dimensional form they read U = 1 6 y Θ y 2 + U y y. (14) Equation (14) again is a good representation of the experimental data. The temperature and velocity profiles, (7) and (12), respectively, are valid for equilibrium boundary layers and for all angles α (angle beteen the all and a horizontal line) ith 3 α < 9. For α 9 only the component of the gravitational acceleration parallel to the all has to be taken into account (g x = g sinα). For α (Rayleigh-Bénard convection) alternative temperature and velocity profiles can be derived, see e.g. Hölling and Herig [11, 12]. 4. Implementation in CFD codes The temperature and velocity profiles derived above can be used as all functions for natural convection in CFD codes, so that coarse grids can be used near the all. Then, the flo and temperature fields are not resolved up to the all, but the center of the first control volume lies ithin the overlap layer. The region close to the all is brigded by the all functions, i.e. gradients at the all are determined using the universal profiles (7) and (12). First, results for natural convection from a commercial CFD-code are shon to emphasize the need for a ne all treatment. In the subsequent section results are given that ere obtained ith the ne all functions. 4

5 8 7 (13) (15) 6 U y Figure 4: Non-dimensional velocity profiles of Betts and Bokhari [2] for Gr = ( ), from Ampofo and Karayiannis [5] for Gr = ( ), of Tsuji and Nagano [3] for Gr L = ( ), for Gr L = ( ), for Gr L = ( ), for Gr L = ( ), for Gr L = ( ), for Gr L = (+) and of Cheeseright [1] for Gr L = ( ). Profiles are shifted by five units each for better clarity. 4.1 Commercial Code Results FLUENT 6.2 as chosen to test the performance of a state-of-the-art commercial CFD-code for turbulent natural convection. The DNS results of Versteegh and Nieustadt [1] served as a benchmark case. The geometry is a heated vertical infinite channel (see Figure 1, left) ith Ra = Gr Pr = and Pr =.7. To different grid sizes ere used, i.e. a coarse grid ith FLUENT standard all treatment and a fine grid resolving the viscous sublayer (y + 1). Additionally, four different (to-equation) turbulence models ere tested on both grids. Wall gradients ere chosen to assess the performance of various combinations since they are equivalent to the all heat flux and all shear stress, both important quantities. In the numerical results temperature all gradients deviate by 2 4%, velocity all gradients by 2 7%. Best results ere obtained ith the k-ω turbulence model for hich the corresponding profiles are shon in Figure 5. In vie of the errors in temperature and velocity gradients and deviations especially in the velocity profiles it can be concluded that commercial codes ith standard turbulence modeling and all treatment give only poor results for turbulent natural convection. 4.2 Ne Wall Functions In order to improve the insufficient results of standard models for turbulent natural convection our ne all functions ere implemented in the OpenSource CFD-code CAFFA [13] ith the k-ω turbulence model and the Boussinesq approximation to account for buoyancy effects. By means of equation (7) and (12) all gradients for the temperature and velocity are calculated, respectively. For simplicity, the boundary condition for k (turbulent kinetic energy) is k = u 2 τ/.3, like in forced convection. The boundary condition for ω (dissipation rate), hoever, as 5

6 T in K u in m/s Figure 5: FLUENT results ( : fine grid; : coarse grid) ith the k-ω turbulence model compared to the DNS data of Versteegh and Nieustadt [1] for Ra = Gr Pr = ( ) T in K u in m/s Figure 6: CAFFA results ( : eight cells; : four cells) using ne all functions compared to the DNS data of Versteegh and Nieustadt [1] for Ra = Gr Pr = ( ). modified. Taking advantage of ν t = k/ω and ν t = ν σ t y Pr C, (15) according to equation (11), the boundary condition for ω is ω = k ν Pr C σ t y. (16) 6

7 k in m 2 /s at/a y/h Figure 7: CAFFA results for the turbulent kinetic energy k (, left) and the ratio of turbulent and molecular thermal diffusivity, a t /a, (, right) compared to the DNS data of Versteegh and Nieustadt [1] for Ra = ( ). The proposed set of all functions / boundary conditions can be used hen the center of the first control volume is ithin the overlap layer (y 5). In Figure 6 results for the infinite vertical channel (Ra = ) obtained ith the ne all functions and modified turbulence boundary conditions are shon. Deviations of the all gradients ith respect to DNS results are ithin 3 % compared to 2 7 % of the FLUENT results. Also, the computation time can be reduced considerably due to the coarse grids that can no be used. Figure 7 shos the profile of turbulent kinetic energy k and the ratio of turbulent and molecular thermal diffusivities, a t /a, compared to the DNS data [1]. Though k deviates from the DNS data, a t /a is quite close to the data in [1]. This is due to the modified boundary condition for ω, equation (16), by hich the correct value of ν t (and therefore also a t ) in the control volume adjacent to the all is enforced. The good agreement of the mean flo field and the rather poor representation of k and ω are not unique for natural convection but can also be observed in forced convection. Numerical solutions for complex interior forced convection flos, as measured by Mocikat et al. [14], for example, sho good agreement for the mean flo profiles based on standard CFD code results. Hoever, e.g. the turbulent kinetic energy k then is rong by more than 1 %. The proposed set of all functions / boundary conditions together ith a standard turbulence model may be a reasonable approach as long as there is no better turbulence model for natural convection. 5. Conclusions The temperature and velocity profiles for turbulent natural convection at vertical alls ere analysed asymptotically. A to-layer structure can be identified ith a viscosity influenced inner layer and a turbulent outer layer. The temperature profile is obtained by asymptotic matching of temperature gradients in the overlap layer. The momentum equation is reritten in terms of the all distance and can then be integrated to get the velocity profile for the overlap layer. These profiles, (7) and (12), are in good agreement ith numerical and experimental data sets. These profiles can be used in CFD-codes as improved all functions for natural convection since commercial codes ith standard all treatment give only poor results. The standard k-ω turbulence model as chosen ith a modified boundary condition for ω. Results for the 7

8 temperature and velocity profiles are in good agreement ith DNS data, though the profile for the turbulent kinetic energy k sho considerable deviations. An alternative turbulence model adapted to natural convection is needed to improve the agreement for k (and ω). Also, an extension to mixed convection at vertical alls is planned, see for example Balaji et al. [15]. Additionally, an extension to non-equilibrium boundary layers ould be an option to increase the range of validity. References [1] T.A.M. Versteegh and F.T.M. Nieustadt, A direct numerical simulation of natural convection beteen to infinite vertical differentially heated alls: scaling las and all functions, Int. J. Heat Mass Transfer 42, (1999). [2] P.L. Betts and I.H. Bokhari, Experiments on turbulent natural convection in an enclosed tall cavity, Int. J. Heat Fluid Flo 21, (2). [3] T. Tsuji and Y. Nagano, Characteristics of a turbulent natural convection boundary layer along a vertical flat plate, Int. J. Heat Mass Transfer 31, (1988). [4] T. Tsuji and Y. Nagano, Turbulence measurements in a natural convection boundary layer along a vertical flat plate, Int. J. Heat Mass Transfer 31, (1988). [5] F. Ampofo and T.G. Karayiannis, Experimental benchmark data for turbulent natural convection in an air filled square cavity, Int. J. Heat Mass Transfer 46, (23). [6] W.K. George and S.P. Capp, A theory for natural convection turbulent boundary layers next to heated vertical surfaces, Int. J. Heat Mass Transfer 22, (1979). [7] R.A.W.M. Henkes and C.J. Hoogendoorn, Numerical determination of all functions for the turbulent natural convection boundary layer, Int. J. Heat Mass Transfer 33, (199). [8] X. Yuan, A. Moser and P. Suter, Wall functions for numerical simulation of turbulent natural convection along vertical plates, Int. J. Heat Mass Transfer 36, (1993). [9] M. Hölling and H. Herig, Asymptotic analysis of the near all region of turbulent natural convection flos, J. Fluid Mech. 541, (25). [1] R. Cheeseright, Turbulent natural convection from a vertical plane surface, J. Heat Transfer 9, 1 8 (1968). [11] M. Hölling and H. Herig, A ne Nusselt/Rayleigh number correlation and all functions for turbulent Rayleigh-Bénard convection, Proc. IHTC13 Sydney, August, (26). [12] M. Hölling and H. Herig, Asymptotic analysis of heat transfer in turbulent Rayleigh- Bénard convection, Int. J. Heat Mass Transfer, 49, (26). [13] J.H. Ferziger and M. Peric, Computational methods for fluid dynamics, 2nd ed., Springer, Berlin (1999). [14] H. Mocikat, T. Gürtler and H. Herig, Laser Doppler velocimetry measurements in an interior flo test facility: A database for CFD-code evaluation, Exp. in Fluids 34, (23). [15] C. Balaji, M. Hölling and H. Herig, Ne approaches to treatment of turbulent mixed convection flos using asymptotics, submitted for publication (26). 8