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Reynolds Averaged Navier Stokes (RANS)

V1. 5

  • Equation for ns changed(Removed time)
  • Changes are bolded

Before Computational Smooth Dynamics(CFD) originated, theoretical studies on high swirling confined turbulent flows can only just be validated by doing experimental studies. These experimental studies require long leading time and high cost. Now, with the aid of CFD, researchers have the ability to study these complicated moves in a much shorter time and with a lower cost incurred.

Many experimental studies have been conducted on the high swirling confined turbulent flows but little has been done on the computational modelling. Many of these intricate movement simulations are completed at the trouble of high computational cost methods such as Large Eddy Simulations(LES) and Direct Numerical Simulations(DNS). Thus, less computational cost alternative will be very useful in the studies of high swirling confined turbulent flows.

Thus, this task will be using the Reynolds Averaged Navier Stokes (RANS) structured turbulence models in ANSYS FLUENT to simulate the high swirling confined turbulent moves in two different test circumstances and the results validated with experimental data. The seeks and targets are discussed as follows:

Aims and Objectives


To validate the correctness of RANS structured turbulence models for the simulation of high swirling restricted turbulent moves.


  1. To simulate the high swirling limited turbulent moves using ANSYS FLUENT with different RANS turbulence models.
  2. To compare the numerical data from the simulations with the experimental data to validate the accuracy of the turbulence models.
  3. To understand the result of the RANS turbulence models on the predicted results.

Review of Limited Swirling Flows

Confined swirling stream takes on an important role in various engineering fields. For instance, they can boost the mixing process in the stirred tanks, enhance the separation of debris in cyclones [1] and also increases the flame stableness in gas turbine combustors. So, exactly what is a swirling circulation?

A swirling flow is a move in which a swirl speed that prevails in the tangential direction apart from the flow movement in the axial and radial guidelines. The swirl velocity of the stream plays a major role in the evolution and decay process of swirling flow motion but not the radial speed of the flow as shown in a study by Beaubert et al. [2]

A swirling move consist of two types of rotational action. A good body rotation at the interior region near to the centerline and a free of charge vortex movement at the exterior region. [14] Sound body rotation and free vortex movement respectively has its velocity straight and inversely proportional to the radius of the pipe at the centre with their axis of rotation as shown in Number 1.

Figure 1: Velocity account of swirling stream in a pipe. [4]

Confined swirling stream may then be grouped into "subcritical" and "supercritical" flows. A "subcritical" circulation has a opposite move at the exit and is also very delicate towards changes at the exit as shown experimentally by Escudier and Keller[11]. Alternatively, the "supercritical" circulation has no opposite movement at the exit and is insensitive towards variant at the exit. [10] "Subcritical" moves are shaped when the ratio of maximum swirl speed to the averaged axial velocity surpasses unity was explained in a theory by Squire[12].

Review of Computational Smooth Dynamics(CFD)

CFD is a strategy which is utilized to study liquid circulation using numerical analysis and algorithms to solve the governing circulation equations. In the past, the field of fluid dynamics was made up of purely experimental and theoretical studies. CFD is considered the "third strategy" in the studies of smooth mechanics and would enhance both existing methods. [5]

The three main elements when implementing CFD will be the pre-processor, solver and post-processor. The pre-processor's task is to convert the input of a movement problem into an application that is suitable for the solver. During pre-processing, the geometry of the situation is defined and the movement domain is split into smaller cells (meshing). The physical (eg: turbulence) and chemical type phenomena that needs to be modelled are chosen and the smooth properties are identified. Next, the boundary conditions receive to cells which interacts with the domain boundary. The perfect solution is to the circulation problem is stored in the nodes in each cell. In the solver, the conservation formula made up of the mass, momentum, energy and varieties is included over each cells. Then, the undiscovered parameters of the formula are interpolated and substituted back into the equation. The solver then operates numerical techniques to solve the derivatives and flux in the cells. Lastly, the post-processor allows end user to analyse the info obtained by plotting graphs and take notice of the flow computer animation. [6]

Review of Turbulence Flows

All essential fluids in action are governed by the conservation of mass formula and the Navier-Stokes equation. The latter equation relates the stream properties like the velocity, pressure, denseness and temp for a moving substance. The conservation of mass equation and the incompressible Navier-Stokes equation (in Cartesian tensor notation) can be respectively written as

Turbulence is shown to develop as an instability in the laminar movement through detailed analysis of the solutions for the Navier Stokes equation. [7].

In principle, Immediate Numerical Simulation(DNS) can be used to simulate very accurate turbulent stream by solving the exact equations with the appropriate boundary conditions. However, it requires large amount of computational electric power as this technique has to signify all of the eddies from the smallest scale to the major scale and the time step chosen must be small enough to solve the quickest fluctuations. The turbulent eddies will be discussed in more detail in the next section.

The two other methods you can use to simulate the turbulent flows (with decreasing computational electric power and accuracy) is the Large Eddy Simulation(LES) and turbulence modelling with Reynold's Averaged Navier Stokes equation (RANS). Fundamentally, LES solves the governing equations partially as only the large eddies are solved using the governing equations and the filtered smaller eddies are modeled while RANS models the whole turbulence eddies in support of the mean parameters are computed.

For turbulence modelling, when details of the turbulent movement are not prioritized so only the common stream properties are resolved. In a very turbulent stream, the speed field fluctuates randomly in both space and time. Regardless of the fluctuations, the time averaged velocity can be motivated and the velocity field equation can be written as:


where is the time averaged velocity and is the fluctuating aspect in the velocity field. Other than the speed, other movement properties may also be decomposed into its mean and fluctuating parts. Inside our simulations, the flow is assumed to be stable, have constant density and axially symmetric. Thus, the incompressible Reynold's Averaged Navier-Stokes (RANS) equations (in Cartesian tensor notation) can be written as

Where is the Reynold's Stress tensor, which is a component of a symmetric second order tensor from the averaged process. The diagonal terms are normal strains while the non-diagonal terms are shear tensions. The Reynolds Stress can be understood as the web momentum transfer anticipated to velocity fluctuations. This term also provided unknown conditions to be equation and thus, more equations need to be found to match the amount of unknowns to resolve the equations.

A straightforward method of "generating" equations would be to create new sets of incomplete differential equations (PDEs) for each and every term using the initial set of Navier-Stokes equation. This can be done by multiplying the incompressible NS equations by the fluctuating property and time averaging them to create the Reynolds-Stress equation. By deriving the Reynolds Stress term, we can identify what's influencing the strain term but the condition with this process is that more unknowns and correlations were made and no new equations are created to account for these unknowns. [7] Thus, these unknown terms have to be modelled to close the formula before they could be used.

Review of Turbulence Eddies

The speed field fluctuations in the turbulence moves are in fact the eddies in the move. The eddies moving go away an object generates the turbulence kinetic energy and the distance range of the eddies, are dependant on the diameter of the object. As the top eddy breakdown into smaller eddies, the turbulence kinetic energy will be passed on and finally dissipated anticipated to viscous forces in the flow. Thus, in line with the Kolmogrov scales, the space and time range of the tiniest eddies depends on the pace they acquire energy from the bigger eddies, and the kinematic viscosity, . It is also noted that the speed of turbulence energy received is equal to the pace of turbulence energy dissipated so, . The Kolmogrov scales shows the distance and time range of the smallest eddies to be and respectively. [8] These expressions may then be used to determine the span and time scale ratio between the small and large eddies.



From the equations above, we can conclude that the top eddies are several purchases of magnitude larger than the tiny eddies. Thus, even at a minimal Reynold's number, enough time and length proportion between your small and large eddies are significant enough to influence the amount of elements and time step required to model the complete turbulent move. Therefore, rather than solving all the eddies, turbulence modelling is required to decrease the amount of computational cost of CFD.


The knowledge of the motions of confined swirling moves and characteristic of the "subcritical" and "supercritical" flows will be useful when describing the simulation results. Before the simulation results are obtained, additionally it is important to identify the basic steps of working any CFD simulations which are the preprocessing, dealing with and post processing.

DNS solves the exact NS formula while LES solves the formula for bigger eddies and models the smaller eddies. The procedure of solving the precise equations occupies a great deal of computational electricity as it would need to represent the all the turbulent eddies engaged and a suitable time step must be chosen to resolve the fluctuations. In comparison with DNS and LES, RANS turbulence modelling requires the least computational power as it does not solve the precise NS equations but instead, models the complete turbulence eddy in support of solves the mean average factors. The low computational cost of RANS turbulence modeling is the principal reason this job has chosen it to simulate the restricted swirling moves. However, the correctness of the methods requires validation, which is the aim of this job.

The RANS turbulence models created depends on the PDEs of the Reynolds stress as a guide as it shows the way the Reynold stress behave. Thus, the next section will sophisticated more about the RANS turbulence models that will be applied in this project.

The main goal of the RAN established turbulence models are to model the (Reynold's Stress tensor) and provide closure to the RANS equation. The three main categories of the turbulence models are linear eddy viscosity models, non-linear viscosity models and Reynolds' Stress Model(RSM). [9]

There are three types of linear eddy viscosity models: algebraic models, one equation models and two formula models. They are based on the Boussinesq hypothesis which models the Reynold's stress tensor to be proportional to the mean rate of pressure tensor, by way of a coefficient known as the eddy viscosity, . This infers that the turbulence circulation field acts similarly to a laminar move field. [10]


The second term of the right hands part of the formula above is necessary when resolving turbulence models that needs to determine the turbulent kinetic energy, k from the travel equations. The formula for k is half the track of the Reynolds Stress tensor.

For the algebraic turbulence models, no additional PDE equations are manufactured to spell it out the carry of the turbulent flux and the alternatives are calculated straight from the movement parameters. An algebraic relation is employed as closure based on the mixing duration theory. The mixing length theory state governments that the eddy viscosity have to vary with the length from the wall. However, the problem with these equations are that they don't account for the effects of turbulence background. In order to increase the turbulent movement predictions, yet another transport equation for k is solved which will replace the velocity scale and include the consequences of turbulence stream history.

For one and two-equation models, the modeled k equation is included thus debate on the precise k equation will first be done. The exact k formula is a PDE derived by multiplying the incompressible NS equations with, averaging it and multiply with. The precise k PDE equation obtained is

The left palm side(LHS) terms will be the material derivative of k which gives the rate of change of turbulent kinetic energy. The first term on the right hand side(RHS) is the development term and represents the turbulent kinetic energy an eddy will gain due to the mean flow tension rate. The second term on the RHS represents the dissipation term which recommended the rate of which the kinetic energy of the smallest turbulent eddy being transferred into thermal energy due to the work done by the fluctuating stress rate contrary to the fluctuating viscous strains. The third term on the RHS is the diffusion term which represents the diffusion of turbulent energy by molecular movement. The past term of the RHS is the pressure-strain term which implies the tendency to redistribute the kinetic energy in the circulation because of the turbulent and pressure fluctuations. In order to close and solve the k equation, the Reynolds Stress, dissipation, diffusion and pressure-strain term needs to be specified.

For the Reynolds Stress term, it has already been mentioned at the beginning that it is based on the Boussinesq hypothesis. The eddy viscosity, is modelled similarly to how it was done for the algebraic models

Where is a constant, the length level of turbulence eddies, is similar the mixing period and velocity scale of the turbulence eddies is changed by the square base of the turbulence kinetic energy, k. The formula above is an isotropic relation which means that it is assumed that the momentum carry is the same in all path at any point.

Next, the dissipation term is modelled predicated on the assumption that the rate of turbulence energy received is add up to the pace of turbulence energy dissipated. Thus, we can write the equation

and since the equation is homogenous, it can be characterized by the space and velocity scale of turbulence eddies giving

Where is a constant.

For the diffusion and pressure-strain term, the sum is modelled based on the gradient diffusion transport mechanism as there is the pressure-strain term is small for incompressible moves. The gradient carry mechanism means that there is a flux of k down the gradient. It is to help ensure that the alternatives are soft and a boundary condition can be applied on k when k is in the boundary. There is absolutely no Therefore, the formula shows

Where is the turbulent Prandtl number and is generally add up to one.

-not completed

- will speak about the modeled turbulent kinetic energy in one equation spalart allmaras

-will speak about dissipation part for 2 formula model in k-e

This test circumstance is chosen because the stream was mapped and documented in detail as So et al could measure and document the flow at length using a Laser Doppler Velocimetry(LDV) at 10 axial stations up to 40d downstream. Thus, the validation of the accuracy of the RANS turbulence models on restricted high swirling move can be carried out.

Description of Test Case

The flow consists of an annular high swirling stream projected into a tube of uniform radius, R = 62. 5mm with a central non-swirling jet of diameter, d = 8. 7mm. The swirl number, S of the stream is determined with

Where U is the axial velocity and W is the swirl velocity. The swirl amount just downstream of the swirl generator is approximately 2. 25 which suggests that it's a high swirling flow and can cause a detrimental pressure gradient at the centreline. The purpose of the non-swirling jet was to postpone the incident of reverse circulation due to the negative pressure gradient along the centreline from 12d to 40d downstream from the inlet.

Geometry (Computational Area)

The restricted swirling flow in this case is a "subcitical" flow according to the guideline of Squire brought up in Section x. Thus, two different computational domains were used for the simulation of the stream to check if the leave geometry will have an impact on the swirling stream simulated.

Figure 2 (short term figure)

The first computational area is the entire geometry of the tube which consist of the computational inlet at x/d =1 and the constriction of 0. 75R from x/d = 70 to the computational wall plug at x/d = 90. The second computational domains is a "cut" faraway from the first domains at x/d = 55 where in fact the constriction is removed.


-have not completed it. Will probably be updated in the next revision.

Boundary Conditions


The inlet experimental measurements for the axial and tangential velocity and stresses are provided. However, the radial velocity component had not been measured and is set to 0 rad/s. The radial stress is also not assessed and was established add up to the tangential stress, whereas the three shear stresses are assumed to be zero. < graphs of recommended to be added>


Conditions at the outlet are not known prior to resolving the circulation problem. No conditions are identified at the outflow boundaries as ANSYS FLUENT will extrapolate the mandatory information from the inside. It is assumed that the flow is fully developed at the leave end thus the outflow boundary condition is used. (dphi/dx|exit = 0)


The no slip condition is applied.

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