Week 8: Literature review - RANS derivation and analysis

1. AIM:

• Explain the Navier-stokes equation.
• Explain turbulence and its relevance in CFD. 
• Propound upon the idea of Reynolds averaging and its need.
• Derivation of RANS.
• What is turbulent Viscosity? Explain RANS turbulence model.

2. WHAT ARE NAVIER-STOKES EQUATIONS?

Navier stokes equations in fluid flow are highly non-linear partial differential equations that describes the fluid flow. Although keeping it simple, in an intuitive sense is Newton’s second law as in it balances the forces which act on a fluid. As shown below.

Now considering volumetric force,

Now here,

So now,

Similarly we can show for the left hand side force term:

Here we will ignore the forces marked in red as they are not always very dominant or considerable. But they will come into effect when there are different cases we consider such as turbo-machinery, Electro-magneto hydrodynamic, etc…

Here we will also make an assumption for the simplicity of derivation that the flow is incompressible and viscous, so we have

Where the second term is viscous stress tensor term and first term is pressure term, it can also be written as:

μ is called the first coefficient of viscosity  and λ is the second coefficient of viscosity or volume viscosity . The value of λ, which produces a viscous effect associated with volume change, is very difficult to determine, not even its sign is known with absolute certainty. Term involving λ is often negligible; though when it is assumed the most common approximation is λ ≈ - ⅔ μ.

From left to right: the first term is transient term which is usually neglected in steady state simulations, second term represents convective transport. The third term is pressure gradient term which captures the effect of pressure and finally fourth term is diffusion term which is effect of viscosity in flow and last S represents source terms like Coriolis force, Gravity and others.

There are lot of other forms of NS equations for different transported variables but in general this form holds true. In case of compressible flows, the density is also coupled with velocity to capture the effects of change in density during high Mach flows.

3. WHAT IS REYNOLDS AVERAGING AND WHY DO WE NEED IT?

As shown above, NS equations CAN be termed as universal transport equation. But these equations are highly non-linear PDEs which is very difficult if not impossible to solve analytically for most fluid cases. Closest we can solve these equation are with various numerical methods most accurate would be DNS. But such solutions become computationally super intensive; in fact most usual fluid flow cases can’t even be solved in a normal office computer with DNS. This required some clever methods to get the solution of NS equation with low computational costs and just enough accurate (resolution) to make engineering decisions.

This is where the averaging comes into picture. Before discussing averaging though, one needs to understand the turbulence.

Big whorls have little whorls, which feed on their velocity, and little whorls have lesser whorls. And so on to viscosity.
– Lewis F. Richardson, 1920

WHAT IS TURBULENCE?

Turbulence is a fluid flow characterised by highly chaotic pressure and velocity changes. It is exactly opposite of Laminar flow where flow takes place with steady shear layers. Turbulence is caused when the flow has excessive kinetic energy which can’t be damped by the viscous effects among the fluid molecules. Turbulence is literally everywhere around us, image below shows a tap with water flowing at different velocities and we can instantly see turbulent flow.

This can be both good and bad. When we want to design a cooling mechanism the turbulence increases the effect heat transfer coefficient. Contrary to that when we want to design aerofoil, turbulence can actually increase drag and reduce lift which is undesirable.

The flow type whether turbulent or laminar is usually discerned by computing Reynolds number as below. It is shown for flow through pipe but different flow geometries would have different Reynolds number ranges which can be determined once experimentally and then can be used to discern the type of flow.

An intuitive way to think about turbulence is to visualize a flow with lots of different scales of eddies and time. Some eddies are large to the size of few meters and some are down to micro-meters. In terms of turbulent eddy scales it is better to visualize them as an energy cascade largest (inertial scale) to smallest (Kolmogorov scale) length scales. Similarly some eddies are moving faster and others move slower, which represents time scales. So it is clear that there will be some spatial resolution required to capture the length scales of eddies as well as some temporal resolution to capture the motion and development of those eddies.

WHAT HAS THAT GOT TO DO WITH NS EQUATIONS BEING HARDER TO SOLVE AND NEED OF RANS?

This is the root of problem in solving NS equations even with DNS. In order to solve the flow completely we need to have extremely refined discretized domain (read: mesh) in order to capture those micro scale eddies perfectly. Besides this, we also need appropriate time resolution (read: time step) to capture the whirls with smallest time steps which makes the computation even more intensive especially if the flow is convection dominant which makes NS PDEs even more non-linear and harder to solve.

In order to counter this, we came up novel approach of ‘averaging’ the NS equation on temporal scale and ‘modelling’ the turbulence length scales. This is what we called as RANS. This produces an averaged results but in most physical systems an engineer can do away with a rough average results than estimating the complete system accurately which would be computationally very intensive not to mention un-necessary in most cases.

Above are some RANS results vs DNS simulations of a fluid jet in open domain. As one can see, DNS resolves all the turbulent and time scales. One can even see some Kevin-Helmholtz instability being captured. In RANS the solution is spatially and temporally averaged and hence it is quite diffused as one can see. Although some important features are not captured in RANS simulation, it is million times less intensive to run and get an averaged solution to make engineering decisions.

4. REYNOLDS AVERAGE NAVIER-STOKES EQUATIONS DERIVATION:

In order to derive RANS form from general equation from, we are considering dynamic viscosity µ to be isotropic in the medium (which strictly speaking is not 100% true).

So we reduce it to the form:

Now here,

To explain averaging look at the following graph, it represents variation of velocity in one dimension and its variation with time in other dimension.

Here we will split the components of velocity in averaged component and fluctuating component.

Overbar represents timestep averaged quantities, hat represents fluctuating component around that average in timestep.

We have taken an unsteady case here which varies with time as one can see even on a larger time scale of tn, but in general the variations are averaged on a small turbulent time scales of T. Usually when running a transient simulation we wouldn’t want to factor T in our simulation as it would require a computation per time step, we would rather take time step such as tn.

Here the averaged velocity  is unsteady over such a large step of tn although minor turbulent scales of time T are averaged and their unsteady component  can be considered as averaged constant. So here the first temporal term would become:

NOTE: here we will only show calculation for the x component and rest shall be inferred from the same following the same process

Similarly if we proceed for 2^nd term, the averaging of convective term will yield as follows

Now here any fluctuating component’s derivative alone would be zero unless they are coupled together in form like:`tildev`.`tildev` So here we finally get.

Next, for the pressure gradient term we can define as:

And last, for the diffusive term we get:

So finally combining all the above equation for the X component we get.

Re-arranging it slightly we get:

Similarly for Y and Z component we can write,

Now adding all the above equations we get:

As shown above, after averaging we get a general RANS from with an additional stress tensor called as Reynolds stress tensor as shown. Notice in more formulations you will see a presence of density there, we have assumed volumetric quantities and that is why the rho is missing.

Similarly we can also show for continuity equation:

5. HOW DOES TURBULENCE MODELS FIT THIS PICTURE? WHAT IS EDDY VISCOSITY?:

The Reynolds-averaged approach to solving a CFD simulation relies on the face that the Reynolds stresses in Equation above have to be appropriately modelled. A common method employs the Boussinesq hypothesis to relate the Reynolds stresses to the mean velocity gradients:

Immediately one can realize almost all terms are readily available except the eddy viscosity or turbulent viscosity `mu_t`.

In order to compute , we employ certain well known turbulence models to estimate it. This essentially leads to the ‘RANS closure’.

Some of the well-known models that use Boussinesq approach:

• K-Epsilon
• K-Omega
• Spalart-Allmaras

The alternative approach to Boussinesq based models is Reynolds Stress Model. This solves transport equations for each of the terms in the Reynolds stress tensor. An additional scale-determining equation (normally for ) is also required. So 5 additional transport equations are required in 2D cases and 7 additional transport equations must be solved in 3D.

Normally RANS based approach is computationally economical and pretty accurate for most type of flows but special cases would require RSM to get physical and accurate results. These flows include high swirling flows, highly anisotropic viscosity in domain and shear dominated flows.

LES deals in a slightly different approach in which large eddies are resolved in time-dependent simulation using low-pass filtering of NS equations above. The idea is that since most large eddies are resolved, the error is in general reduced in the simulation. For the remaining smaller scales of the turbulence, it is easier to find a general model since at such regions in domain the viscosity is highly isotropic. But for most flows encountered in real life, LES still remains a computationally extremely expensive choice.  This is mainly because, although not as much as DNS, but LES still requires very small spatial and temporal resolution.

WHAT IS TURBULENT VISCOSITY? HOW IS IT DIFFERENT?

As we discussed above, eddy viscosity models rely on something called Turbulent Viscosity.  Simply put when flow is laminar, diffusion is dominated by intermolecular force called viscosity (molecular). But when the flow becomes turbulent, diffusion happens not only by molecular forces, but also by turbulent eddies.

This leads to additional momentum diffusion compared to just the viscosity model alone. This viscosity which is additional is termed as turbulent or eddy viscosity.

By second law of thermodynamics, molecular viscosity has to be positive but since the turbulent viscosity is a concept it can be positive as well as negative.