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Viscous Stress Terms for the RELAP5-3D Momentum Equations

Adam Kraus and George Mesina

RELAP5 International Users Seminar 2010September 20-23, 2010

Outline of Presentation• Motivation for calculating viscous stresses• Previous work on liquid phase (Raymond Wang, Berkeley)• Compressibility and other changes for gas• Discretizations, boundary conditions• Discussion of results for laminar flow• Preliminary turbulent flow results• Future work

Motivation for Viscous Stress Implementation

• Improved physical modeling• Greater accuracy for laminar flow regimes• Second-order effect

– Accuracy– Improves convergence time

• Better coupling of RELAP5 to CFD codes– Provide more accurate inlet flow field to CFD code

Previous Work (viscous)• Liquid phase viscous terms implemented

• Cylindrical and Cartesian coordinates

• Applies only to:– Laminar flow regimes– Incompressible fluids– Semi-implicit time advancement

• Performed by Raymond Wang (Berkeley)• Subroutine “viscous” in flux3d

Gas Phase Viscous Calculations (viscousG)• Compressibility

– Additional terms in Navier-Stokes Equations (NSE)– Usually ignored if Mach < 0.3– No “bulk viscosity” term

• Radial Velocity NSE

• Incompressible• Compressible

Momentum Control Volume Grid (R-θ)

Momentum Control Volume Grid (R-Z)

Compressible Term Discretizations• Discretized with central differences

• Example 1-direction discretization (pure partial derivative):

• Example 2-direction discretization (mixed partial):

Subroutine for Gas Equation Viscous Terms• “No-slip” boundary condition (for viscous terms only)• Special handling developed for singularity at r=0 in cylindrical

• Implemented in a new subroutine called “viscousG” – Corrects a couple programming errors in found viscous– Implements incompressible viscous term discretization with new

naming convention– Implements compressibility term discretization – Implements boundary condiitons and r=0 handling

Test Model 1: Poiseuille Flow in Cylindrical Pipe

• Inlet velocity profile: either plug or Poiseuille• Uniform mesh for each coordinate direction• 25 axial zones (L=2500m)• 9 radial zones (R=1m)• 1 azimuthal zone

Liquid Data Error norms (%) L1 L2 L∞ Original RELAP5

652.07 362.76 342.27

viscous 14.00 5.50 3.12 viscousG 7.87 2.79 1.55

, H2O, 800°C Gas Data Error Norms (%) Inlet type/ Axial level

L1 L2 L∞

Pois/ z = 22 9.85 4.16 3.26 Pois/ z = 12 11.71 4.75 3.37 Plug/ z = 22 11.46 4.44 2.62 Plug/ z = 12 13.11 5.12 2.73

Test Model 2: Rectangular Duct • Inlet velocity profile: either plug or parabolic• Uniform mesh for each coordinate direction• 5 x-zones, 5 y-zones (width for both is 5m)• 5 axial zones (L=500m)

Rectangular Flow Results

• Much more qualitative

Other Factors Affecting Velocity Profile• Significant Effects

– Wall friction– Misapplied boundary condition (potentially)

• Minor Effects– Change of fluid– Compressibility– “Numerical viscosity”

Turbulent Stress Modeling• RANS (general form)

• represents previously calculated viscous terms (in tensor form)• is known as the Reynolds-stress tensor (must be modeled)

• Selected turbulence model: Prandtl mixing length model

where • U is the mean axial velocity, y is the distance from the boundary wall

• Discretization:

Liquid Data Error norms (%) L1 L2 L∞ Original RELAP5

103.82 52.75 43.83

Mixing length

102.24 46.80 35.71

Turbulent Stress Modeling

• Modeling does not greatly reduce error of results

• Better profile shape, follows power law

• Problems must be run at very small time steps

• Other differencing techniques should be investigated for better stability

Further Work

• Apply only on wall cells in viscousG

• Extend current work to nearly-implicit time advancement

• Increase the maximum number of radial mesh increments (currently 9)

• Stability analysis of numerical methods for turbulence calculations

• Investigate more complex turbulence models

Acknowledgements

• My mentor, Dr. George Mesina

• The entire RELAP5 team

• Idaho National Laboratory and the DOE Office of Science