A numerical analysis is performed using ANSYSCFX to investigate backward-facing step flow for Reynolds numbers in the turbulent regions. Reattachment lengths are determined for each Reynolds number.
Flow over a backward-facing step produces recirculation zones where the fluid separates and forms vortices. For turbulent flow, the fluid separates at the step and reattaches downstream, as show below in figure 1. Only a single recirculation zone develops for turbulent flow and the reattachment point is believed to be independent of the Reynolds number and depend only on the ratio of inlet height to outlet height.
The aims of this report are to:
- Create a backward facing step simulation appropriate for turbulence modelling.
- Determine the effects the different meshing schemes have on the final outcome.
- Simulate turbulent flow and compare with the benchmark data.
- Determine the effects of different turbulence models.
- Understand the relevant flow characteristics.
MATHEMATICAL FORMULATION FOR TURBULENT FLOW
The governing equations for computational fluid dynamics (CFD) are based on conservation of mass, momentum, and energy
The basic equations for steady-state laminar flow are conservation of mass and momentum. When heat transfer or compressibility is involved the energy equation is also required. The governing equations are,
Continuity Equation:
Momentum equation:
where, τ, the stress tensor is,
Turbulent flow can be modeled using mean and fluctuating values for components, such as velocity, . Substituting the mean and fluctuating value equations into the Navier-Stokes equations yields the Reynolds-averaged Navier-Stokes (RANS) equations:
The k-ε model is semi-empirical two-equation turbulence model that is based on an exact solution for the turbulent kinetic energy (k) and a model of the dissipation rate (ε). To model the Reynolds stress, , in the RANS equations,
the κ-ε model uses the Bousinesq approximation to relate the Reynolds stresses to the mean velocity gradients.
Along with the Bousinesq approximation above, the following definition of the eddy viscosity is used,
The realizable portion of the k-ε model is based on the following relationship, which can be obtained by determining the point that the average normal stress becomes negative. The realizable k-ε model coefficient,, is determined by equilibrium analysis at high Reynolds numbers.
The realizable k-ε model is defined by the following two equations,
and
Gk represents the generation of turbulence kinetic energy due to the mean velocity gradients and relies on the Boussinesq approximation.
where the modulus of the mean rate-of-strain tensor,
and
The variable in the eddy viscosity is,
where,
and the model constants are,
where,
The following values are used for the remaining constants,
MODEL DEFINITION AND GENERATION
Geometry
The flow boundary of the model was created as we are modeling flow over a body. This was done so by using ANSYS Workbench which represents the flow domain as specified in the assignment handout. The geometry of the 2D Duct was cut out of as specified.
Figure 1 – Schematic of backward-facing step turbulent-flow
The geometry for the backward-facing step used in this analysis is as below
FIGRE 2
Fig: 2D Duct model
Using ANSYS WORKBENCH
SPECIFICATIONS AND BOUNDARY CONDITIONS
The Reynolds number is defined as, , where u is the inlet velocity, υ is the kinematic viscosity, and D is the hydraulic diameter. Here it is given as Reh= 5100(Constant).
Corresponding to this Reynolds number the velocity will be 0.1333 m/s.
Miscellaneous reference values
Governing equations
• Mass continuity for incompressible flow
∇*U = 0
• Steady flow momentum equation
∇* (UU) + ∇*R = −∇p
where p is kinematic pressure and (in slightly over-simplistic terms)R = ν ∇U is the viscous stress term with an effective kinematic viscosity ν ,calculated from selected transport and turbulence models.
Initial conditions U = 0 m/s, p = 0 Pa
Boundary conditions
• Inlet (left) with fixed velocity u = (0.1333, 0, ) m/s;
• Outlet (right) with fixed pressure p = 0 Pa;
• No-slip walls on other boundaries.
Transport properties
• Kinematic viscosity of air ν = 15.68 μm2/s
Turbulence model
• Standard k − epsilon;
Coefficients: Cμ = 0.09;C1 = 1.44;C2 = 1.92; αk = 1; αǫ = 0.76923.
MESHING
The combination of robust and automated surface, inflation and tetrahedra meshing using default physics controls to ensure a high-quality mesh suitable for the defined simulation allows for push-button meshing. Local control for sizing, matching, mapping, virtual topology, pinch and other controls provide additional flexibility, if needed.
A mesh of sufficient quality was generated ensuring that aspect ratio was less than 4 the mesh was found to be within desired limits. As the name suggests skewness represents how deformed the mesh is, having high skewness would result in errors in results as viscous effects would create incorrect results and pass this information onto the next control volume hence amplifying the error.
Uniform structured quadrilateral mesh
Non-uniform structured quadrilateral mesh with refinement in appropriate regions
Refinement at lines
Refinement at areas
Unstructured triangular mesh with refinement in appropriate regions
Best quality mesh
Non-uniform structured quadrilateral mesh with refinement
Necessity of mesh refinement
The primary benefit of these techniques is their increased efficiency that results from meshing, i.e. from the improvement of the existing mesh which more efficiently represents the pertinent parts of the problem.
In numerical analysis, mesh refinement is a method of meshing. Central to any Eulerian method is the manner in which it discretizes the continuous domain of interest into a grid of many individual elements. This grid may be static, established once and for all at the beginning of the computation, or it may be dynamic, tracking the features of the result as the computation progresses. If the computation has features which one wants to track which are much smaller than the overall scale of
the problem, and which move in time, then one must either include many more static grids to cover the region of interest, or adopt a dynamic scheme.
The advantages of a dynamic gridding scheme are:
- Increased computational savings over a static grid approach.
- Increased storage savings over a static grid approach.
- Complete control of grid resolution, compared to the fixed resolution of a static grid approach, or the Lagrangian-based adaptivity of smoothed particle hydrodynamics.
A STANDARD K-ε TURBULENCE MODEL
Turbulent flow over a backward facing step is a standard test case used to compare and validate turbulence models. The case presented here, Figure 2, uses a domain in which the channel height is five times the height of the step, h. The inflow boundary is positioned one step heights upstream of the step to limit the effect of the assumptions made at the inflow boundary on the solution in the region after the step.
At this boundary the turbulent quantities are estimated using the formulae
The inflow velocity, u is chosen to give a Reynolds number of 5100. The outflow boundary is placed four step heights downstream of the step, again to to limit its influence on the simulation. The reattachment length, xR, is a commonly used parameter to determine the ability of a turbulence model to correctly simulate the backward facing step. This parameter is the distance from the step to the position on the wall, at the bottom of the channel, at which the velocity along the channel becomes positive.
Below Figure shows a plot of the velocity component along the channel in the line of elements immediately above the channel wall.
A REALIZABLE K-ε TURBULENCE MODEL
One of the more successful recent developments is the realizable K-Epsilon model developed by Shih et al. This model contains a new transport equation for the turbulent dissipation rate . Also, a critical coefficient of the model, , is expressed as a function of mean flow and turbulence properties, rather than assumed to be constant as in the standard model. This allows the model to satisfy certain mathematical constraints on the normal stresses consistent with the physics of turbulence (realizability). The concept of a variable is also consistent with experimental observations in boundary layers.
The backward facing step problem was simulated using the RNG variation of the k - epsilon model. For this case the graph of the velocity component along the channel, in the line of elements immediately above the lower wall, is plotted in Figure below. The reattachment length in this case is slightly closer to the experimental results than the standard k - epsilon model but is still a 12.3% error. In nearly all cases the RNG k - epsilon model produces a slightly better result than the standard k - epsilon model, but the errors in the RNG model are still of the same order as the errors in the results from the standard k - epsilon model.
Best Model:
Realizable k-ε turbulence model
The realizable K-Epsilon model is substantially better than the standard K-Epsilon model for many applications, and can generally be relied upon to give answers that are at least as accurate. Both the standard and realizable models have been implemented in ANSYS CFX with a two-layer approach, which enables them to be used with fine meshes that resolve the viscous sublayer.
Realizable k-ε turbulence model for Ethylene Glycol
Realizable k-ε turbulence model for Ethylene Glycol at 37°C at the same Reynolds number (Reh= 5100).
When we run the simulation for the above stated fluid material we get the same properties as was get in the air flow .The only difference is that the Realizable k-ε turbulence model in this case have a little bit difference in reattachment length of approximate of 0.10 % .Therefore we can say that the model simulation for ethylene glycol flow and air flow is same and having all the properties same.
RESULTS AND DISCUSSIONS
Turbulent flow
A plot of the streamlines over the step is displayed in figure below. A second eddy near the step corner is observed. The velocity of the recirculation zone is on the order of magnitude lower than the velocity at the step.
Solving for turbulent flow required multiple levels of refinement to obtain an accurate solution.
A general trend in the turbulent results of this analysis is slightly lower values for separation and reattachment points than compared with other numerical studies. This difference with present results for turbulent flow can be attributed to the range of methods and grids used to perform the numerical calculations.
Stream wise velocity contours with streamlines
Pressure Distribution along length
Turbulent kinetic energy with streamlines
CONCLUSION
The flow over a backward facing step is often used as a test case for analysing the performance of computational methods and turbulence models. It embodies several important aspects of turbulent flow, flow separation, recirculation and reattachment.
The use of ANSYS CFX as a demonstration tool for the study of a fluid mechanics
Phenomenon is proven to be very useful to students in understanding the case of viscous flow reattachment in duct. Such case of flow separation cannot be represented by ready-to-use equation, and neither can it be easily observed through simple experiment. Thus with the availability of the CFD computational software like ANSYS CFX, difficult-to-imagine cases like the flow separation could be better understood by students at undergraduate level. The study enables students to learn the concept of flow separation from a different view, besides getting the idea that the occurrence of flow separation is inter-related with the shape or bending radius of the elbow.
In this study the standard k-ε turbulence model is not better than realizable k-ε turbulence model and we also see that the turbulent flow for backward step modelling is provides better understanding about the effects of different turbulence models and the relevant flow characteristics.
