CFD calculation of convective heat transfer coefficients and validation Laminar and Turbulent flow Subtask 3: Boundary Conditions by Adam Neale M.A.Sc. Student Concordia University Dominique Derome Concordia University Bert Blocken K.U.Leuven Jan Carmeliet K.U.Leuven/Technical University of Eindhoven
CFD Calculation of Surface Coefficients Part 1: Laminar Flow - Simulation of Forced Convection Part 2a: Turbulent Flow - Simulation of Forced Convection Wall boundaries Wall boundary - Analytical Validation - Semi-empirical Validation Part 2b: Turbulent Flow - Simulation of Natural Convection - Validation case for wall functions
CFD Calculation of Surface Coefficients Part 1 Laminar Flow Cases Computational Domain : Y X Height = 0.05 m Length = 3.0 m Boundary Conditions : Inlet Conditions U av = 0.1 m/s U(y) = 3/2*U av *[1 4*(y/b) 2 ] m/s T = 283 K Wall boundaries Case (A): Constant Heat Flux (CHF) q w = 10 W/m 2 Case (B): Constant Wall Temperature (CWT) T w = 293K q w = 10 W/m 2 T w = 293K
CFD Calculation of Surface Coefficients Part 1 Laminar Flow Mesh 19800 cells b = 0.05m L = 3m
CFD Calculation of Surface Coefficients Part 1 Laminar Flow Boundary Conditions (B.C.) Case A (CHF): Velocity Inlet B.C. U av = 0.1 m/s U(y) = 3/2*U av *[1 4*(y/b) 2 ] m/s T = 283 K Wall B.C.: q w = 10 W/m 2 Pressure Outlet B.C. Wall B.C.: q w = 10 W/m 2 Case B (CWT): Velocity Inlet B.C. U av = 0.1 m/s U(y) = 3/2*U av *[1 4*(y/b) 2 ] m/s T = 283 K Wall B.C.: T w = 293K Wall B.C.: T w = 293K Pressure Outlet B.C.
CFD Calculation of Surface Coefficients Part 1 Laminar Flow Case A, Constant Heat Flux: h c values 4 h c (W/m 2 K) 3.5 3 2.5 2 1.5 T b n i= = 1 ( u b T ) U i av i b i h cref hc-ref h cc hcc h cb hcb Analytical 1 0.5 0 0 0.5 1 1.5 2 2.5 3 X Position (m)
CFD Calculation of Surface Coefficients Part 1 Laminar Flow Case B, Constant Wall Temperature: h c values 4 h c (W/m 2 K) 3.5 3 2.5 2 1.5 h cref hc-ref h cc hcc hcb h cb Analytical 1 0.5 0 0 0.5 1 1.5 2 2.5 3 X Position (m)
CFD Calculation of Surface Coefficients Part 1 Laminar Flow Grid sensitivity analysis φ h φ 2h φ 4h φ 8h φ 16h Number of cells in the Y Direction Number of cells in the X Direction Smallest cell height (m) Smallest cell width (m) Total number of cells 67 33 17 8 4 1200 600 300 150 75 4.202E-04 8.749E-04 1.775E-03 3.948E-03 9.147E-03 0.0025 0.005 0.01 0.02 0.04 80400 19800 5100 1200 300 *Initial mesh
CFD Calculation of Surface Coefficients Part 1 Laminar Flow Richardson Extrapolation Using h c at 2.5m as a reference (i.e. φ h = h c ) Discretization error: ε d h φh 2 φ a 2h 1 Order of the scheme: a = φ2h φ4 log φh φ2h log ( 2) h Richardson grid independent solution: d Φ = φ h + ε h Finest mesh solution
CFD Calculation of Surface Coefficients Part 1 Laminar Flow Case A: Grid sensitivity results 2.02 9.0 Φ = 1.992875 1.97 1.975 1.987 1.991 8.0 7.0 h c (W/m 2 K) 1.92 1.931 x=2.5m Richardson Relative error 6.0 5.0 4.0 3.0 Relative Error (%) 1.87 1.848 2.0 1.0 1.82 φ16h (300) φ8h (1200) φ4h (5100) φ2h (19800) φh (80400) 0.0 Grid (#cells)
CFD Calculation of Surface Coefficients Part 1 Laminar Flow Case B: Grid sensitivity results 1.85 9.0 Φ = 1.82509 1.80 1.812 1.821 1.824 8.0 7.0 h c (W/m 2 K) 1.75 1.769 x=2.5m Richardson Relative error 6.0 5.0 4.0 3.0 Relative Error (%) 1.70 1.690 2.0 1.0 1.65 φ16h (300) φ8h (1200) φ4h (5100) φ2h (19800) φh (80400) 0.0 Grid (#cells)
CFD Calculation of Surface Coefficients Part 1 Laminar Flow Results Order of the scheme a d ε h Discretization Error (W/m 2 K) Finest mesh solution (W/m 2 K) Richardson Solution (W/m 2 K) Analytical solution h c (W/m 2 K) CHF 1.460 2.297x10-3 1.990578 1.992875 1.992875 φh Φ CWT 1.858 1.001x10-3 1.824089 1.825090 1.824922
CFD Calculation of Surface Coefficients Part 2a Turbulent Flow Part 2a: Turbulent Flow - Simulation of Forced Convection Part 2b: Turbulent Flow - Simulation of Natural Convection - (In)Validation of wall functions Wall boundary - Semi-empirical Validation
CFD Calculation of Surface Coefficients Part 2a Turbulent Flow Universal Law-of-the-wall u+ 25 20 15 10 u+ Equation (7) u+ Equation (10) u+ Spalding Equation (13) T+ Equation (8) T+ Equation (11) 25 20 15 10 T+ 5 5 0 1 10 100 1000 0 y+
CFD Calculation of Surface Coefficients Part 2a Turbulent Flow Guidelines for near-wall modelling Height of first cell above the wall 1) Laminar Sublayer (y + < 5) Low-Reynolds number modelling 25 20 15 u+ Equation (7) u+ Equation (10) u+ Spalding Equation (13) T+ Equation (8) T+ Equation (11) 25 20 15 u+ T+ 2) Buffer Region (5 < y + < 30) 10 10 Not recommended 5 5 0 0 1 10 100 1000 3) Log-law Layer (30 < y + < 60) y+ Wall functions
CFD Calculation of Surface Coefficients Part 2a Turbulent Flow Computational Domain Symmetry BC U = 0.5m/s T = 283 K Y Velocity Inlet BC Pressure Outlet BC H = 1m Wall BC - Constant Heat Flux X q w = 10 W/m L = 5m Note : Not to scale.
CFD Calculation of Surface Coefficients Part 2a Turbulent Flow Meshes y + 1 50,000 cells y + 30 1300 cells
CFD Calculation of Surface Coefficients Part 2a Turbulent Flow Simulations Low-Re Simulations 1. Spalart-Allmaras Model 2. Standard k-ε Model 3. RNG k-ε Model 4. Realizable k-ε Model 5. Standard k-ω Model 6. SST k-ω Model 7. Reynold s Stress Model (RSM) Wall Function Simulations 1. Standard k-ε Model 2. Standard k-ω Model
CFD Calculation of Surface Coefficients Part 2a Turbulent Flow u + Results 25 u+ 20 15 Semi-Empirical Equation - Laminar Sublayer Semi-empirical Equation - Log-law Empirical Equation - Spalding (1961) k-e standard k-e RNG k-e realizable k-w standard k-w SST Spalart-Allmaras RSM WF - ke WF - kw 10 5 0 1 10 100 1000 y+
CFD Calculation of Surface Coefficients Part 2a Turbulent Flow T + Results 16 T+ 12 8 Semi-empirical Equation - Laminar Sublayer Semi-empirical Equation - Log-law k-e standard k-e RNG k-e realizable k-w standard k-w SST Spalart-Allmaras RSM WF - ke WF - kw 4 0 1 10 100 1000 y+
CFD Calculation of Surface Coefficients Part 2a Turbulent Flow h c Results 6 h c (W/m 2 K) 5 4 3 k-e standard k-e RNG k-e realizable k-w standard k-w SST Spalart-Allmaras RSM k-e WF k-w WF Lienhard (2006) Eq. 6.111 Lienhard (2006) Eq. 6.115 2 1 0 0 1 2 3 4 5 X Position (m)
CFD Calculation of Surface Coefficients Part 2b Turbulent Flow Methodology for validation Part 2a: Turbulent Flow - Simulation of Forced Convection Part 2b: Turbulent Flow - Simulation of Natural Convection - (In)Validation of wall functions Wall boundary - Semi-empirical Validation
CFD Calculation of Surface Coefficients Part 2b Natural Convection (In)Validity of Wall Functions y = 3m Air, T initial = 293K Isothermal Hot Side y = 2.25m Cold Side T w1 = 298K y = 1.50m T w2 = 288K Y y = 0.75m 0m 0m X Isothermal X = 3m Note : Not to scale.
CFD Calculation of Surface Coefficients Part 2b Natural Convection Meshes y + 2 (22500 cells) y + 30 (900 cells)
CFD Calculation of Surface Coefficients Part 2b Natural Convection Velocity profiles 0.8 0.7 Low-Re Velocity profile at y = 2.25 m, Hot side Spalding wall function equation 0.6 Velocity (m/s) 0.5 0.4 0.3 0.2 0.1 0 0 0.05 0.1 0.15 Position (m)
CFD Calculation of Surface Coefficients Part 2b Natural Convection Velocity fields m/s Low-Re Modelling Wall Function
CFD Calculation of Surface Coefficients Part 2b Natural Convection Dimensionless velocity 30 25 20 Low-Re: y=0.75m, Hot side Low-Re: y=1.50m, Hot side Low-Re: y=2.25m, Hot side WF: y=0.75m, Hot Side WF: y=1.50m, Hot Side WF: y=2.25m, Hot Side Spalding Wall Function Equation Low-Re: y=0.75m, Cold side Low-Re: y=1.50m, Cold side Low-Re: y=2.25m, Cold side WF: y=0.75m, Cold Side WF: y=1.50m, Cold Side WF: y=2.25m, Cold Side u+ 15 10 5 0 1 10 100 1000 10000 y+