Help with Understanding Governing Equations of CFD

[m.gos]

I need some help in understanding some of the governing equations of computational fluid dynamics. I already have some books to read, others are on their way, yet I still find it difficult because I learn much better on examples rather than pure theory. I was hoping someone could 'read out' for me the equations below in a semi-mathematician semi-fluid-dynamicist manner, if you know what I mean.
For instance I know what $$\nabla$$ is (in cartesian coordinates), but I still have not figured out how does $$\nabla\bullet$$V (V is vector velocity field in cartesian space) differ from V$$\bullet\nabla$$. I also know local derivatives, substantial derivatives (tota derivative with respect to time), but when it comes to reading equations with understanding I get confused.
First set of equations refers to mass transfer (continuity equation) and is as follows:
http://img855.imageshack.us/i/chds.png/
Second set of equations refers to RENS (Raynolds-averaged Navier-Stokes) k-e turbulance model, so a bit more complicated:
http://img846.imageshack.us/i/nitf.png/Thank you in advance,
Mat

 

[Studiot]

Hello, m.gos and welcome to PF. Heady stuff !

First the difference between v dot nabla and nabla dot v.

In the following I have replaced v with $$\nu$$ to avoid confusion with their components.

Let $$\nu$$ be the velocity with components u, v, w. relative to the x, y, z axes.

Let there also be a function f(x,y,z) of some quantity of interest (eg density) with respect to these same axes

$$\nu$$ dot nabla is used with a function to display the spatial variation of that function.

nabla dot $$\nu$$ is the spatial variation of the velocity itself.

Specifically, when written out in full.

$$\nu .\nabla f = u\frac{{\partial f}}{{\partial x}} + v\frac{{\partial f}}{{\partial y}} + w\frac{{\partial f}}{{\partial z}}$$

It can be seen that nabla is really operating on f to create a vector which is then dot (pre)multiplied by $$\nu$$

and

$$\nabla .\nu = \frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} + \frac{{\partial w}}{{\partial z}}$$

I note in your attachments you refer to D which is a function peculiar to fluid dynamics. D refers to the rate of change 'following the fluid'. That is it tracks some quantity of interest of a particular parcel of fluid as it moves from point to point.
Use of D normally involves introducing time, which I have not done above for simplicity.

Does this help?

 

[m.gos]

Thanks a lot Studiot.

Even though you focused on my first question you gave to it the precise answer I needed. Looks very sensible once you know it. Also thanks for the symbol explenation.
I think I should also give some more comment regarding the second part of my post where I ask how to read the equations from the two links. First of all here are some symbols definitions:

• c is the concentration of the species (mol/m3)
• D denotes the diffusion coefficient (m2/s)
• R is a reaction rate expression for the species (mol/(m3·s))
• u is the velocity vector (m/s)
• R is a reaction rate expression for the species (mol/(m3·s))
• N is an arbitrary user-specified flux expression (SI unit: mol/(m2·s)).
• $$\rho$$ is the density (SI unit: kg/m3)
• Q contains heat sources other than viscous heating (SI unit: W/m3)
• T is absolute temperature (SI unit: K)

All I really need are the first two equataions from each set, I am sure this will be enough for me to do the rest myself. Now, to show what I am after I will give it a try with the 2nd equation from the second set:
The equation is a partial differential form of the continuity (mass conservation) equation. It states that for the considered element of fluid the sum of the time rate of change of density at the fixed point in space (local derivative) and the time rate of change of (mass flow?) equals zero. Does it make sense? Can someone do a similar thing with other 3 equations?

Cheers

 

[Studiot]

Well I didn't realize you were coming from a chem eng direction.

Fluids books seem to cost extra - an arm and a leg, but Chem eng books always cost yet more - two arms and two legs - you have my sympathies.

Since this is about transport phenomenona, my comment about D was misplaced. Thanks you for the key to the symbols - makes life a deal easier.

My D is really an operator and used thus:

$$\frac{{Df}}{{Dt}} = \frac{d}{{dt}}f\left\{ {x\left( t \right),y\left( t \right),z\left( t \right),t,} \right\} = \frac{{\partial f}}{{\partial x}}\frac{{dx}}{{dt}} + \frac{{\partial f}}{{\partial y}}\frac{{dy}}{{dt}} + \frac{{\partial f}}{{\partial x}}\frac{{dz}}{{dz}} + \frac{{\partial f}}{{\partial t}}$$

$$= \frac{{\partial f}}{{\partial t}} + u\frac{{\partial f}}{{\partial x}} + v\frac{{\partial f}}{{\partial y}} + w\frac{{\partial f}}{{\partial x}}$$

$$= \frac{{\partial f}}{{\partial t}} + \nu .\nabla f$$

Which was one of your original enquiries.

If no one has helped further I will try to look tomorrow, but it is after midnight here.

go well

 

[blue_raver22]

thias

Sure, I would be happy to help you understand these governing equations of computational fluid dynamics (CFD). CFD is a complex field that combines principles from mathematics, physics, and engineering to simulate and study fluid flow. Understanding the governing equations is crucial for accurately modeling and predicting fluid behavior.

Let's start with the first set of equations, which is the continuity equation. This equation describes the conservation of mass in a fluid flow. In other words, it states that the rate of change of mass in a given volume is equal to the net flow of mass into or out of that volume. The equation is written as:

∂ρ/∂t + ∇•(ρV) = 0

In this equation, ρ represents the density of the fluid, t is time, V is the velocity vector field, and ∇• represents the divergence operator. This operator essentially measures the rate at which a quantity (in this case, mass) is flowing out of a given point in space. This equation is often written in Cartesian coordinates, but it can also be written in other coordinate systems.

Now, let's move on to the second set of equations, which is the Reynolds-averaged Navier-Stokes (RANS) equations with the k-ε turbulence model. The Navier-Stokes equations are the fundamental equations of fluid dynamics and describe the motion of viscous fluids. In CFD, these equations are solved numerically to simulate fluid flow.

The RANS equations are a modified version of the Navier-Stokes equations that take into account the effects of turbulence. Turbulence is a chaotic and random motion of fluid particles, and it is important to consider in many practical applications. The k-ε turbulence model is a commonly used approach to model turbulence and is based on two equations: one for the turbulent kinetic energy (k) and one for the rate of dissipation of this energy (ε).

The RANS equations with the k-ε turbulence model are written as:

∂(ρV)/∂t + ∇•(ρV⃗⃗) = -∇p + ∇•(μ(∇V + (∇V)^T)) + ρg + ∇•(ρkV)

∂k/∂t + V•∇k = Pk - ε + ∇•(μ(∇V + (∇V)^T))