In CFD, it is considered that the computation of any problem solving, starts with grid generation, but actually it begins with the governing equations of fluid dynamics. Everything in CFD is based on the continuity, momentum and energy equation that serves as fundamental governing equations of fluid dynamics.
Before starting with the conservation equations, we have to be clear about some mathematical tools to understand. Sorry, not only to understand but to admire the real beauty of fundamental governing partial differential equations of fluid dynamics.
There are three major tools namely,
- Models of the flow
- Substantial derivative
- Divergence of the velocity
In this article, we’ll be discussing more about the “Models of the Flow”.
Before that here is a question! Why we are more concerned about the Models of the Flow & why is it necessary to study about the models of the flow before knowing the governing equations? Let’s explore.
Models of the Flow:
If we consider a solid body which is in motion, it is indeed easy to see and define it, because it is a rigid body, because all the particles in the solid body travel at same velocity. But the same does not apply for the motion of fluid , and it is not that easy to define it even though you could see it (water or some liquid),because it’s a squishy substance that is very hard to hold of. And also the velocity of each part of the fluid body may be different at each location in the fluid. But that squishy substance obeys to the physical principles. Hence how do we visualize the moving fluid, to apply physical principles? For that we take some assumptions to construct the models of the flow to prove the physical principles mathematically.
Here we will simulate four kinds of fluid flow models by basically considering two approaches, namely Finite Control Volume approach and Infinitesimal Fluid Element approach.
Finite Control Volume (FCV):
Consider a flow field and imagine a closed finite region of the flow, that constitutes a volume called “Control Volume V” and a closed surface which bounds the volume called “Control Surface S”. Figure is shown for both V and S.
Out of four models two are from FCV. They basically differ with the motion of the CV.The CV may be fixed in space, with the fluid moving through it. Figure 1 may give an overview for above mentioned lines.
The CV may be moving with the fluid, same fluid particles are always inside it. Figure 2 may give an overview for above mentioned lines.
The above two mentioned are the models of the flow FCV. Actually in either case, the CV is a reasonably large, finite region of the flow. Here the fundamental physical principles are applied to the fluid inside the control volume and to the fluid crossing the control surface (if it is fixed in space). Hence we can limit our attention towards CV instead of looking at the whole flow field. The fluid flow equations that we can directly obtain by applying the fundamental physical principles to a finite control volume are in integral form. With some manipulation, we obtain the PDE from integral form of the governing equations.
We could name the integral or PDE equation for the above two models of the FCV. The equation obtained from the FCV which is fixed in space is called Conservation form.
The equation obtained from the FCV which is moving along with the fluid is called Non-conservation form. But we could take some assumptions in the above forms.
In the fixed FCV in space, the Control Volume will be constant because it is fixed in space, so the fluid is will flow through it. Hence the volume will not change.If the control volume is constant, obviously control surface is also constant. But the mass of the fluid will vary accordingly.
Likewise in the moving FCV with the fluid, control volume will vary because some parts of fluid which is taken as Control Volume moves. Therefore to state the properties of flow it is necessary that we should rely on to the same fluid particles. In the moving parts of the fluid (control volume), not all the fluid particles travels with the same velocity. Some moves in random directions with different velocity. So in that case we could assume that the mass of the fluid will be constant but control volume and surface varies.
Infinitesimal Fluid Element:
Consider a general flow field as represented by the streamlines .Let us imagine an infinitesimally small element in the flow with a differential volume dV. The fluid element is infinitesimal in the same sense as differential calculus. The previous statement means that the fluid element is taken as very small.
Here a question may even arise in your mind, that how much small is volume of the fluid element?
Actually in the microscopic approach, there might be a certain maximum range to consider. Likewise, in the macroscopic approach, certain minimum range to consider. In between both levels, the volume of the fluid element lies because the finite control volume can be imagined as some volume which could have some reasonable value or magnitude. When compared to FCV, the IFE is very small and sometimes negligible. But it is large enough to contain a huge number of molecules, so that it can be viewed as a continuous medium.
Like FCV, fluid element may be fixed in space with the fluid moving through it called conservation form and fluid element moving along a streamline equal to flow velocity at each point called non-conservation form.
Again, instead of taking whole flow field, the fundamental physical principles are applied to infinitesimal small fluid element. But this applies directly to the PDE form like finite control volume.
The models of the flow are a kind of tool which is very helpful in converting the physical principle into mathematical expressions or statements.
Well. That’s it about the Models of flow and very soon you will get to know about Substantial derivative and divergence of the velocity in the upcoming blogs.