Friction Factor Estimation for Turbulent Flow in Corrugated Pipes - - PowerPoint PPT Presentation

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

1

Problem Setting

2

Two-Equation Turbulence Models and BCs Reynolds Averaging, k − ǫ and k − ω Models. Law of the Wall

3

Turbulent Flow in Conventional Pipes Smooth Wall Case Rough Wall Case

4

Friction Factor Computations

Problem Setting Two-Equation Turbulence Models and BCs Turbulent Flow in Conventional Pipes Friction Factor Computations Summary

Where We Are Now

1

Problem Setting

2

Two-Equation Turbulence Models and BCs Reynolds Averaging, k − ǫ and k − ω Models. Law of the Wall

3

Turbulent Flow in Conventional Pipes Smooth Wall Case Rough Wall Case

4

Friction Factor Computations

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

1

Problem Setting

2

Two-Equation Turbulence Models and BCs Reynolds Averaging, k − ǫ and k − ω Models. Law of the Wall

3

Turbulent Flow in Conventional Pipes Smooth Wall Case Rough Wall Case

4

Friction Factor Computations

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: ∂˜ uj ∂xj = 0 ← continuity equation ρ ∂˜ ui ∂t + ˜ uj ∂˜ ui ∂xj

• = − ∂˜

p ∂xi + ∂˜ T(v)

ij

∂xj ← NS equation ˜ T(v)

ij

• stress due to viscous forces

Newtonian fluid hypothesis: ˜ T(v)

ij

= µ ∂˜ ui ∂xj + ∂˜ uj ∂xi

• .

Problem Setting Two-Equation Turbulence Models and BCs Turbulent Flow in Conventional Pipes Friction Factor Computations Summary

Reynolds Averaging

Reynolds decomposition: ˜ ui = Ui + ui, ˜ p = P + p, ˜ T(v)

ij

= T(v)

ij

+ τ (v)

ij ,

Ui, P, T(v)

ij

• mean components; ui, p, τ (v)

ij

• fluctuating components.

∂Uj ∂xj = 0. ρ ∂Ui ∂t + Uj ∂Ui ∂xj

• = − ∂P

∂xi + ∂ ∂xj [T(v)

ij

− ρ uiuj Reynolds stress ]. ρ uiuj 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: σij ≡ −ρ uiuj = µT ∂Ui ∂xj + ∂Uj ∂xi

• µT - "turbulence viscosity" (eddy viscosity), [N/m2·s] (not constant).

k - specific turbulence kinetic energy, [N·m/kg=m2/s2]. ǫ - turbulence dissipation, [m2/s3]. ω - turbulence dissipation per unit turbulence kinetic energy, [1/s]. µT = ρCµ k2 ǫ µT = 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 N(˜ ui) = ρ∂˜ ui ∂t + ρ˜ uk ∂˜ ui ∂xk + ∂˜ p ∂xi − µ∂2˜ ui ∂x2

k

, Take the following moment of NS operator uiN(˜ uj) + ujN(˜ ui) = 0 ⇒ equation for ρuiuj. Turbulence kinetic energy (per unit mass) k ≡ 1 2 uiui = 1 2

• u2

1

• +
• u2

2

• +
• u2

3

• .

Take the trace of the Reynolds stress equation ρ∂k ∂t + ρUj ∂k ∂xj = σij ∂Ui ∂xj − ρǫ + ∂ ∂xj

• (µ + µT

σk ) ∂k ∂xj

• .

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 2µ ρ ∂ui ∂xj ∂N(ui) ∂xj

• = 0 ⇒ equation for ǫ = µ

ρ ∂ui ∂xj ∂ui ∂xj

• .

ρ∂ǫ ∂t + ρUj ∂ǫ ∂xj = Cǫ1 ǫ kσij ∂Ui ∂xj − Cǫ2ρǫ2 k + ∂ ∂xj

• µ + µT

σǫ ∂ǫ ∂xj

• .

Cǫ1, Cǫ2, σǫ - modeling constants. ρ∂ω ∂t + ρUj ∂ω ∂xj = αω k σij ∂Ui ∂xj − βρω2 + ∂ ∂xj

• (µ + σωµT) ∂ω

∂xj

• .

α„ β, σω - 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 U+

p = 1

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 U+

p = 1

κ ln y+

p + B,

where U+

p = Up

u∗ , y+

p = u∗yp

ν u∗ = τw ρ , τw is unknown! From Prandtl’s assumption (νT = κu∗y) and the balance of production and dissipation of turbulent kinetic energy: u∗ = C1/4

µ k1/2 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

Up u∗ = 1 κ ln y+

p + B.

Multiply by u2

∗

Upu∗ = u2

∗

1 κ ln y+

p + B

• .

Replace u2

∗ by τw/ρ, and u∗ by C1/4 µ k1/2 p

UpC1/4

µ k1/2 p

= τw ρ 1 κ ln y+

p + B

• .

The skin friction force at the wall (or wall stress), τw: τw = ρC1/4

µ k1/2 p 1 κ ln y+ p + BUp.

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). τw = (µ + µT)∂Up ∂n . (µ + µT)∂Up ∂n − ρC1/4

µ k1/2 p 1 κ ln y+ p + BUp = 0, y+ p = ρC1/4 µ k1/2 p

yp µ . a(kp, ǫp)∂Up ∂n +b(kp)Up = 0. ← Robin BC with variable coefficients Zero-flux BC (no turb. energy transfer through the boundary) n · ∇kp = 0, ǫp = C3/4

µ k3/2 p

κyp , ωp = C−1/4

µ

k1/2

p

κyp yp - 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

1

Problem Setting

2

Two-Equation Turbulence Models and BCs Reynolds Averaging, k − ǫ and k − ω Models. Law of the Wall

3

Turbulent Flow in Conventional Pipes Smooth Wall Case Rough Wall Case

4

Friction Factor Computations

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

Le D ≈ 4.4Re1/6, 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 U(r, 0) = U(r, L) k(r, 0) = k(r, L) ǫ(r, 0) = ǫ(r, L) ω(r, 0) = ω(r, L) P(r, 0) = Pin P(r, L) = Pout Symmetry axis n · ∇U = 0 n · ∇k = 0 n · ∇ǫ = 0 n · ∇ω = 0 Wall (µ + µT)∂Up

∂n = ρC1/4

µ k1/2 p 1 κ ln y+ p +BUp

n · ∇kp = 0 ǫp = C3/4

µ k3/2 p

κyp

, ωp = C−1/4

µ

k1/2

p

κyp

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

Problem Setting Two-Equation Turbulence Models and BCs Turbulent Flow in Conventional Pipes Friction Factor Computations Summary

Law of the Wall for Rough Walls

U u∗ = 1 κ ln yp e

• + 8.5

= 1 κ ln y+

p + [8.5 − 1

κ ln e+] e - roughness height e+ - non-dimensional roughness

Problem Setting Two-Equation Turbulence Models and BCs Turbulent Flow in Conventional Pipes Friction Factor Computations Summary

Law of the Wall for Rough Walls

U u∗ = 1 κ ln yp e

• + 8.5

= 1 κ ln y+

p + [8.5 − 1

κ ln e+] e - roughness height e+ - non-dimensional roughness

Problem Setting Two-Equation Turbulence Models and BCs Turbulent Flow in Conventional Pipes Friction Factor Computations Summary

"Combined" Law of the Wall

Introduce a "combined" law of the wall: U u∗ = 1 κ ln y+

p + B∗,

where B∗ = B + θ(8.5 − B − 1 κ ln e+), Hydrodynamic smoothness - θ = 0 (e+ < e+

1 )

Full roughness - θ = 1 (e+ > e+

2 )

Transition - θ = θ(e+), e+

1 < e+ < e+ 2

θ = sin π 2 ln(e+/e+

1 )

ln(e+

2 /e+ 1 )

• ,

e+

1 = 2.25, e+ 2 = 90.

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

Where We Are Now

1

Problem Setting

2

Two-Equation Turbulence Models and BCs Reynolds Averaging, k − ǫ and k − ω Models. Law of the Wall

3

Turbulent Flow in Conventional Pipes Smooth Wall Case Rough Wall Case

4

Friction Factor Computations

Problem Setting Two-Equation Turbulence Models and BCs Turbulent Flow in Conventional Pipes Friction Factor Computations Summary

Computational Domain and BCs

Pressure across the pipe is not constant in corrugated pipes P(r, 0) = P(r, L) + ∆P

Problem Setting Two-Equation Turbulence Models and BCs Turbulent Flow in Conventional Pipes Friction Factor Computations Summary

Initial Mesh

Problem Setting Two-Equation Turbulence Models and BCs Turbulent Flow in Conventional Pipes Friction Factor Computations Summary

Adapted Mesh

Problem Setting Two-Equation Turbulence Models and BCs Turbulent Flow in Conventional Pipes Friction Factor Computations Summary

Conergence vs. Iteration Number (Adaptive Solver)

Problem Setting Two-Equation Turbulence Models and BCs Turbulent Flow in Conventional Pipes Friction Factor Computations Summary

A Typical Solution

Problem Setting Two-Equation Turbulence Models and BCs Turbulent Flow in Conventional Pipes Friction Factor Computations Summary

Friction Factor vs. Wall Roughness

Problem Setting Two-Equation Turbulence Models and BCs Turbulent Flow in Conventional Pipes Friction Factor Computations Summary

Conclusions

k − ǫ model is slightly better than k − ω model To get a realistic estimation of the friction factor the laws for smooth and rough wall have to be combined The roughness of the fabric has a secondary influence on the friction factor To obtain a considerable decrease in friction factor shape

• ptimisation for the steel spiral should be considered