Problem Setting Two-Equation Turbulence Models and BCs Turbulent Flow in Conventional Pipes Friction Factor Computations Summary Friction Factor Estimation for Turbulent Flow in Corrugated Pipes with Rough Walls Maxim Pisarenco Department of Mathematics and Computer Science Eindhoven University of Technology August, 2007
Problem Setting Two-Equation Turbulence Models and BCs Turbulent Flow in Conventional Pipes Friction Factor Computations Summary Outline Problem Setting 1 Two-Equation Turbulence Models and BCs 2 Reynolds Averaging, k − ǫ and k − ω Models. Law of the Wall Turbulent Flow in Conventional Pipes 3 Smooth Wall Case Rough Wall Case Friction Factor Computations 4
Problem Setting Two-Equation Turbulence Models and BCs Turbulent Flow in Conventional Pipes Friction Factor Computations Summary Where We Are Now Problem Setting 1 Two-Equation Turbulence Models and BCs 2 Reynolds Averaging, k − ǫ and k − ω Models. Law of the Wall Turbulent Flow in Conventional Pipes 3 Smooth Wall Case Rough Wall Case Friction Factor Computations 4
Problem Setting Two-Equation Turbulence Models and BCs Turbulent Flow in Conventional Pipes Friction Factor Computations Summary Flexible Pipes Respond well to bending Easy to install Excellent strength/length ratio Corrugated Rough walls
Problem Setting Two-Equation Turbulence Models and BCs Turbulent Flow in Conventional Pipes Friction Factor Computations Summary Simulation of Turbulent Flows 3 basic approaches: DNS - Direct Numerical Simulation LES - Large-Eddy Simulations RANS - Reynolds-Averaged Navier-Stokes ← DNS solution ← RANS solution
Problem Setting Two-Equation Turbulence Models and BCs Turbulent Flow in Conventional Pipes Friction Factor Computations Summary Where We Are Now Problem Setting 1 Two-Equation Turbulence Models and BCs 2 Reynolds Averaging, k − ǫ and k − ω Models. Law of the Wall Turbulent Flow in Conventional Pipes 3 Smooth Wall Case Rough Wall Case Friction Factor Computations 4
Problem Setting Two-Equation Turbulence Models and BCs Turbulent Flow in Conventional Pipes Friction Factor Computations Summary Equations Describing the Dynamics of Flow Incompressible flow equations: ∂ ˜ u j = 0 ← continuity equation ∂ x j T ( v ) ∂ ˜ � ∂ ˜ � ∂ ˜ = − ∂ ˜ u i u i p ij ρ ∂ t + ˜ u j + ← NS equation ∂ x j ∂ x i ∂ x j T ( v ) ˜ - stress due to viscous forces ij Newtonian fluid hypothesis: � ∂ ˜ � + ∂ ˜ u i u j T ( v ) ˜ = µ . ij ∂ x j ∂ x i
Problem Setting Two-Equation Turbulence Models and BCs Turbulent Flow in Conventional Pipes Friction Factor Computations Summary Reynolds Averaging Reynolds decomposition: ˜ = U i + u i , u i ˜ p = P + p , T ( v ) T ( v ) + τ ( v ) ˜ = ij , ij ij U i , P , T ( v ) - mean components; u i , p , τ ( v ) - fluctuating components. ij ij ∂ U j = 0 . ∂ x j � ∂ U i � ∂ U i = − ∂ P + ∂ [ T ( v ) ρ ∂ t + U j − ρ � u i u j � ] . ij ∂ x j ∂ x i ∂ x j � �� � Reynolds stress ρ � u i u j � is unknown ← closure problem
Problem Setting Two-Equation Turbulence Models and BCs Turbulent Flow in Conventional Pipes Friction Factor Computations Summary Eddy Viscosity Approximation Newtonian type closure, proposed by Boussinesq: � ∂ U i � + ∂ U j σ ij ≡ − ρ � u i u j � = µ T ∂ x j ∂ x i µ T - "turbulence viscosity" (eddy viscosity), [N/m 2 · s] (not constant). k - specific turbulence kinetic energy, [N · m/kg=m 2 /s 2 ]. ǫ - turbulence dissipation, [m 2 /s 3 ]. ω - turbulence dissipation per unit turbulence kinetic energy, [1/s]. k 2 µ T = ρ C µ ǫ µ T = ρ k ω Both modeled on dimensional grounds.
Problem Setting Two-Equation Turbulence Models and BCs Turbulent Flow in Conventional Pipes Friction Factor Computations Summary Turbulence Energy Equation - Outline of Derivation Define the NS operator as − µ∂ 2 ˜ u i ) = ρ∂ ˜ ∂ ˜ + ∂ ˜ u i u i p u i N (˜ ∂ t + ρ ˜ , u k ∂ x 2 ∂ x k ∂ x i k Take the following moment of NS operator � u i N (˜ u j ) + u j N (˜ u i ) � = 0 ⇒ equation for ρ � u i u j � . Turbulence kinetic energy (per unit mass) k ≡ 1 2 � u i u i � = 1 �� � � � � �� u 2 u 2 u 2 + + . 1 2 3 2 Take the trace of the Reynolds stress equation � � ρ∂ k ∂ k ∂ U i − ρǫ + ∂ ( µ + µ T ) ∂ k ∂ t + ρ U j = σ ij . ∂ x j ∂ x j ∂ x j σ k ∂ x j
Problem Setting Two-Equation Turbulence Models and BCs Turbulent Flow in Conventional Pipes Friction Factor Computations Summary Turbulence Dissipation Equation - Outline of Derivation Take the following moment of NS operator � ∂ u i � � ∂ u i � 2 µ ∂ N ( u i ) = 0 ⇒ equation for ǫ = µ ∂ u i . ρ ∂ x j ∂ x j ρ ∂ x j ∂ x j � ∂ǫ �� � − C ǫ 2 ρǫ 2 ρ∂ǫ ∂ǫ ǫ ∂ U i k + ∂ µ + µ T ∂ t + ρ U j = C ǫ 1 k σ ij . ∂ x j ∂ x j ∂ x j σ ǫ ∂ x j C ǫ 1 , C ǫ 2 , σ ǫ - modeling constants. � � ρ∂ω ∂ω = αω ∂ U i − βρω 2 + ∂ ( µ + σ ω µ T ) ∂ω ∂ t + ρ U j k σ ij . ∂ x j ∂ x j ∂ x j ∂ x j α „ β , σ ω - modeling constants.
Problem Setting Two-Equation Turbulence Models and BCs Turbulent Flow in Conventional Pipes Friction Factor Computations Summary Boundary Conditions Law of the Wall p = 1 U + κ ln y + p + B .
Problem Setting Two-Equation Turbulence Models and BCs Turbulent Flow in Conventional Pipes Friction Factor Computations Summary Law of the Wall Law of the Wall p = 1 U + κ ln y + p + B , p = U p p = u ∗ y p where U + , y + ν u ∗ � τ w u ∗ = ρ , τ w is unknown! From Prandtl’s assumption ( ν T = κ u ∗ y ) and the balance of production and dissipation of turbulent kinetic energy: µ k 1 / 2 u ∗ = C 1 / 4 . p
Problem Setting Two-Equation Turbulence Models and BCs Turbulent Flow in Conventional Pipes Friction Factor Computations Summary Law of the Wall as Boundary Condition U p = 1 κ ln y + p + B . u ∗ Multiply by u 2 ∗ � 1 � U p u ∗ = u 2 κ ln y + p + B . ∗ ∗ by τ w /ρ , and u ∗ by C 1 / 4 µ k 1 / 2 Replace u 2 p � 1 � = τ w µ k 1 / 2 U p C 1 / 4 κ ln y + p + B . p ρ The skin friction force at the wall (or wall stress), τ w : τ w = ρ C 1 / 4 µ k 1 / 2 p p + BU p . κ ln y + 1
Problem Setting Two-Equation Turbulence Models and BCs Turbulent Flow in Conventional Pipes Friction Factor Computations Summary Law of the Wall as Boundary Condition (2) The stress at the wall (or anywhere else) can be computed as a function of the velocity gradient (Newtonian Fluid approximation). τ w = ( µ + µ T ) ∂ U p ∂ n . ∂ n − ρ C 1 / 4 µ k 1 / 2 p = ρ C 1 / 4 µ k 1 / 2 ( µ + µ T ) ∂ U p y p p p p + BU p = 0 , y + . κ ln y + 1 µ a ( k p , ǫ p ) ∂ U p ∂ n + b ( k p ) U p = 0 . ← Robin BC with variable coefficients Zero-flux BC (no turb. energy transfer through the boundary) n · ∇ k p = 0 , ǫ p = C 3 / 4 µ k 3 / 2 , ω p = C − 1 / 4 k 1 / 2 p µ p κ y p κ y p y p - distance from the wall, free parameter
Problem Setting Two-Equation Turbulence Models and BCs Turbulent Flow in Conventional Pipes Friction Factor Computations Summary Sensitivity of Solution to the Choice of y + p Solution is not changing for 50 < y + p < 300
Problem Setting Two-Equation Turbulence Models and BCs Turbulent Flow in Conventional Pipes Friction Factor Computations Summary Where We Are Now Problem Setting 1 Two-Equation Turbulence Models and BCs 2 Reynolds Averaging, k − ǫ and k − ω Models. Law of the Wall Turbulent Flow in Conventional Pipes 3 Smooth Wall Case Rough Wall Case Friction Factor Computations 4
Problem Setting Two-Equation Turbulence Models and BCs Turbulent Flow in Conventional Pipes Friction Factor Computations Summary The Moody Diagram
Problem Setting Two-Equation Turbulence Models and BCs Turbulent Flow in Conventional Pipes Friction Factor Computations Summary Fully Developed Flow L e D ≈ 4 . 4 Re 1 / 6 , for turbulent flow
Problem Setting Two-Equation Turbulence Models and BCs Turbulent Flow in Conventional Pipes Friction Factor Computations Summary Computational Domain and BCs Inflow/Outflow Symmetry axis Wall ∂ n = ρ C 1 / 4 µ k 1 / 2 ( µ + µ T ) ∂ U p U ( r , 0 ) = U ( r , L ) n · ∇ U = 0 p p + B U p κ ln y + 1 k ( r , 0 ) = k ( r , L ) n · ∇ k = 0 n · ∇ k p = 0 ǫ ( r , 0 ) = ǫ ( r , L ) n · ∇ ǫ = 0 ǫ p = C 3 / 4 µ k 3 / 2 , ω p = C − 1 / 4 k 1 / 2 p µ p ω ( r , 0 ) = ω ( r , L ) n · ∇ ω = 0 κ y p κ y p P ( r , 0 ) = P in P ( r , L ) = P out
Problem Setting Two-Equation Turbulence Models and BCs Turbulent Flow in Conventional Pipes Friction Factor Computations Summary Meshes and Solution Procedure
Problem Setting Two-Equation Turbulence Models and BCs Turbulent Flow in Conventional Pipes Friction Factor Computations Summary Computed vs. Measured Friction Factor
Problem Setting Two-Equation Turbulence Models and BCs Turbulent Flow in Conventional Pipes Friction Factor Computations Summary Computed vs. Measured Friction Factor
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