Computational Fluid Dynamics For Turbulent Flows

Turbulent flows:  If you have observed smoke rising from a chimney or cigarette, or water stream from a tap, you must have noticed a steady flow followed by a chaotic flow region.

The flow in the chaotic region is referred to as turbulent flow while the steady flow is referred to as laminar flow.

Let’s imagine a flow through an open channel with a constant flow velocity along its height, so that the fluid layers move parallel to one another. This type of flow is purely laminar. Now, if you introduce a small disturbance at the inlet, it will cause the velocity in a few of the layers to change. This results in differential velocities across the layers, which induces a circulation known as eddies, and this phenomenon is called turbulence. But the question arises: How is the flow disturbed in the first place? 

The characteristic of the flow—whether it is laminar or turbulent—does not depend on whether the flow was disturbed, but rather on how the flow responds to any small disturbances that occur in the fluid.

To draw a parallel, consider a guitar string. It doesn’t matter how you pluck the string; it produces the same tone because once plucked, the string vibrates at its natural frequency, which depends on its length, diameter, and material. Similarly, the tendency of the flow to be turbulent, and its intensity, depends on the properties of the flow itself and not on the disturbance. The disturbances we are discussing here are very small in comparison to the overall flow dimensions, but these seemingly insignificant disturbances can alter the flow characteristics depending on the flow properties.

Reynolds Number:

Reynolds number is a unitless quantity that tells us about the tendency of the flow to be laminar or turbulent. 

The formula shows that the value depends on the density, velocity, characteristic length, and viscosity.  

How can we relate all of these quantities to the turbulence?

• Viscosity: By definition, viscosity is a property of the fluid that causes resistance to the relative motion between the fluid layers. Since the circulations are nothing but the effect of the relative motion between fluid flows, viscosity has great significance when discussing turbulent flows.  The highly viscous fluids such as engine oil have so large viscous forces that any small disturbance in the flow will eventually die out due to resistance from neighboring fluid layers.

• Velocity: When the flow has high velocity, any small disturbance in the flow will result in high variations in the velocities normal to the flow direction. If these disturbances have so much energy that they are dominant compared to the viscous forces, the disturbance will move along with the fluid causing circulations. 

• Density: For high-density fluids, if they have a difference in the velocities in a direction normal to the flow, the difference in their momentum will be much higher and hence the viscous forces will not be enough to completely diffuse the energy of the disturbance causing it to continue with the flow.

• Characteristic length: Consider a pipe of a large diameter, since the shear stress in liquid given by, 

The net force acting on the nth layer depends on the du/dy value. For lower du/dy, the resisting viscous force on the nth layer will be much lesser hence there can be an nth layer where the velocities are much different from that of mainstream velocity generating large eddies.

Those who know little about CFD must be aware that there are whole books on turbulence and it is studied as a separate branch in CFD, but why?

If you have noticed from the fundamental equations for solving pressure and velocity fields we have two equations out of which the continuity equation does not have a pressure term. Since we do not have any equation to couple pressure and velocity it makes it impossible to solve these equations.

The way to solve these equations and calculate the pressure and velocity field is by iterative method which can be done using computers where the fluid domain is divided into small control volumes. 

The problem lies in the size of these control volumes because the fluctuations that make the flow turbulent are of such a small size that it is impossible to have a control volume smaller than that to capture the flow irregularities and solve the real-life problem, the available computers do not allow us to solve such a big problem. The method that uses very small control volumes and aims to solve the turbulence problem is known as DNS (Direct Numerical Simulation) and is computationally costly.

To approximate the problem and make it computationally easy to solve scientists have proposed various methods and the whole study of turbulence in commercial problems revolves around these methods which are categorized into two LES and RANS.

LES (Large eddy simulation): 

The large eddy simulation solves the equation like DNS with a good enough size of the control volume which can capture the larger eddies in the flow. The size of our grid or control volume needs to be very very smaller than the size of the eddies we intend to capture.

To solve for the smaller eddies or fluctuations there are analytical equations proposed.

We apply a filter on the Navier-Stokes equation to separate the large-scale

 component and small-scale components. The large-scale components are those responsible for large eddies and are larger than the grid size. The smaller components however cannot be solved using simulation as they are smaller than the grid size and are known as subgrid stresses and they need to be modelled separately.

The velocity and pressure therefore can be given by equations:

u(x,t) = u(x,t) + u'(x,t)  & p(x,t)= p(x,t) + p'(x,t)  

Notice that here, both the large-scale part and small-scale part are time-dependent.

RANS (Reynolds averaged Navier Stokes): 

The RANS is developed based on the fundamental concept of mean flow and perturbations. The mean flow is something that does not change with time while the perturbations or fluctuations in the flow are dynamic and cause the turbulence. 

The velocity and pressure thus can be given by:

u = u + u'(x,t)  & p= p + p'(x,t) .

u  & p  are mean flow properties that are steady.

In most of the practical problems we are only interested in the overall pattern of the flow and not the minute details of the turbulence. Due to the simplicity and improved time complexity, RANS is widely used in industry applications.

The unsolved mystery of turbulence persists despite our limited ability to solve it in practical applications, even with the use of Direct Numerical Simulation (DNS). There are numerous factors contributing to turbulence, such as temperature fluctuations, impurities in the fluid, irregularities in the flow region or walls, fluctuations at the source, and forces arising from other processes like chemical reactions.

While the origin of turbulence might seem insignificant, the effects it produces can be highly significant, depending on flow properties such as the Reynolds number. We are far from solving these problems, especially without a unified theory that accounts for all types of forces. Even if such a theory existed, solving large-scale problems would still pose a considerable challenge, as we lack computers capable of handling such complex simulations.

We need to understand that the current human understanding of the laws of the universe is approximate, whether it pertains to gravity, electromagnetism, chemical processes, or fluid dynamics.

-Pruthvi Utturwar (Research Engineer / Developer at Paanduv Applications)