# Turbulent Flow with Separation in an Axisymmetric Diffuser

In this application, AcuSolve is used to simulate fully developed turbulent flow through an axisymmetric diffuser with a divergent upper wall and a straight lower wall. AcuSolve results are compared with experimental results as described in Driver (1991) and on the NASA Langley Research Center Turbulence Modeling Resource webpage. The close agreement of AcuSolve results with experimental data and reference turbulence model performance validates the ability of AcuSolve to model cases with turbulent flow with separation due to an adverse pressure gradient within an axisymmetric geometry.

## Problem Description

The problem consists of a fluid with material properties close to air flowing around a cylinder through an axisymmetric diffuser, as shown in the following image, which is not drawn to scale. The diffuser has a divergent section designed to generate an adverse pressure gradient. The inlet height between the cylinder and the upper section of the diffuser is 0.0355 m. The diffuser is constructed such that the section is axisymmetric around a cylinder with a diameter of 0.14 m. The inflow of the diffuser is set to produce a fully developed turbulent flow profile at a Reynolds number (Re) of 2,000,000. The upper section is modeled with a slip boundary condition, in order to match experimental conditions. The density of the fluid is 1.0 kg/m3 and with a dynamic viscosity of 1.5 X 10-5 kg/m-s. The simulation was conducted with the Reynolds Averaged Navier-Stokes equations using four turbulence models, Spalart Allmaras, Shear Stress Transport (SST), K-ω and Realizable K-ε.
The problem is simulated as axisymmetric by considering a 1 degree portion of the diffuser with the axisymmetric boundary condition applied on the cross-stream surfaces. The mesh elements were extruded in the axisymmetric direction to create a mesh with two elements spanning the simulated section.

## AcuSolve Results

The AcuSolve solution converged to a steady state and the results reflect the mean flow conditions. As the fully developed turbulent flow enters the divergent section, the expansion of the cross-sectional height causes the streamwise velocity to decrease. This causes separation of the flow along the cylinder wall and results in an area of recirculation that eventually recovers downstream. After the flow enters the diffuser, the velocity decreases significantly from the inlet velocity, due to the rapid expansion of the cross section height in the divergent section. A separation bubble forms within the diffuser, with decreased flow velocity and increased pressure near the lower wall.
Upstream of the diffuser section, the streamwise velocity increases as the distance from the cylinder wall increases, with no influence from the top wall of the diffuser. As the flow enters into the divergent section, the streamwise velocity decreases near the cylinder wall and changes direction in the recirculation region. Within the recirculation region, the reduced speed of the streamwise flow causes the wall shear stress to decrease and eventually change direction for a short distance along the cylinder surface. The images below show the coefficient of pressure (Cp) and coefficient of skin friction (Cf) along the cylinder wall in the diffuser compared against experimental results. The non-dimensional values are defined by the integrated inlet pressure and the magnitude of the inlet velocity. In the images black circles represent the experimental measurements (Driver 1991), solid red lines represent the prediction for the SA model, solid blue lines represent the prediction for the SST model, solid green lines represent the prediction for the K-ω model and solid cyan lines for the K-ε model, representing the AcuSolve results. There are minor differences in the prediction of the pressure coefficient within the diffuser between the three turbulence models. The SA model predicts a slightly higher value of pressure as the flow separates from the cylinder wall compared to both SST and K- ω. The shearing on the cylinder wall shows similar behavior, with the SA model predicting a slightly longer recirculation region and with the K- ω model predicting the separation further upstream. It was found that all three turbulence models predict the separation point further upstream than what is shown in the experiment. This is shown to be consistent with previous studies for each of the turbulence models used (NASA 2014).

## Summary

In this application, a fully developed turbulent flow at a Reynolds number of 2,000,000 is studied and compared against experimental data. The AcuSolve results compare well with the experimental data for pressure coefficient and skin friction coefficient. The recirculation region is slightly over predicted by the turbulence models studied, but the results still show a reasonable trend compared to experimental results. The performance of the three turbulence models was found to be consistent with previously published results for flow within an axisymmetric diffuser (NASA 2014). The results of this validation demonstrate the ability of AcuSolve to accurately predict flow recirculation with axisymmetric boundary conditions.

## Simulation Settings for Turbulent Flow with Separation in an Axisymmetric Diffuser

AcuConsole database file: <your working directory>\axisymmetric_diffuser_turbulent\axisymmetric_diffuser_turbulent.acs

Global

• Problem Description
• Analysis type - Steady State
• Turbulence equation - Spalart Allmaras
• Auto Solution Strategy
• Max time steps - 150
• Relaxation Factor - 0.4
• Material Model
• Fluid
• Density - 1.0 kg/m3
• Viscosity - 1.5e-5 kg/m-sec

Model

• Volume
• Fluid
• Element Set
• Material model - Fluid
• Surfaces
• Axisymmetric_maxZ
• Simple Boundary Condition - (disabled to allow for periodic conditions to be set)
• Axisymmetric_minZ
• Simple Boundary Condition - (disabled to allow for periodic conditions to be set)
• Inlet
• Simple Boundary Condition
• Type - Inflow
• Inflow type - Velocity
• X velocity - 30.0 m/sec
• Turbulence input type - Direct
• Eddy viscosity - 1.5e-5 m2/sec
• Outlet
• Simple Boundary Condition
• Type - Outflow
• Slip_maxY
• Simple Boundary Condition
• Type - Slip
• Wall_minY
• Simple Boundary Condition
• Type - Wall
• Periodics
• Axisymmetric
• Periodic Boundary Condition
• Type - Axisymmetric
• Rotation Axis
• Point 1
• x-coordinate - 1.4 m
• y-coordinate - 0.0 m
• z-coordinate - 0.0 m
• Point 2
• x-coordinate - -1.5 m
• y-coordinate - 0.0 m
• z-coordinate - 0.0 m

## References

D. M. Driver. "Reynolds Shear Stress Measurements in a Separated Boundary Layer Flow". AIAA Paper 91-1787 from the AIAA 22nd Fluid Dynamics, Plasma Dynamics, and Lasers Conference. Honolulu, HI. June 1991.

NASA Langley Research Center Turbulence Modeling Resource webpage, http://turbmodels.larc.nasa.gov/driver_val.html. Accessed December 2014.