Using FTCS to solve a reduced form of Navier-Stokes eqn

DA6righthand · March 30, 2015, 14:57

I'm relatively new to CFD and took on research in CFD. I'm a mechanical engineering major...so I'll do my best in describing the given problem.

The problem regards two parallel plates extended to infinity at a distance of h apart. The fluid between the plates has a given density and kinematic viscosity. The upper plate is stationary and the lower plate is suddenly set in motion with a constant velocity of 40 m/s. The spacing h is 4 cm. A constant streamwise pressure gradient of dp/dx is imposed within the domain at the instant motion starts. A spacial size of 0.001 m is specified. The reduced form of the Navier-Stokes equation is

$\frac{\partial u}{\partial t}=\nu \frac{\partial ^2u}{\partial y^2}-\frac{1}{\rho }\frac{\partial p}{\partial x}$

Using finite differences, I get the following

$\frac{\partial u}{\partial t}\approx \frac{u^{n+1}_{j,k}-u^n_{j,k}}{\Delta t}$
$\frac{\partial ^2u}{\partial y^2}\approx \frac{u^n_{j,k+1}-2u^n_{j,k}+u^n_{j,k-1}}{\Delta y^2}$

I'm confused about the $\frac{\partial p}{\partial x}$ term. I'm supposed to solve for velocity using the FTCS explicit scheme with 1) $\frac{dp}{dx}=0.0$ , 2) $\frac{dp}{dx}=20000.0$ Pa/m, and 3) $\frac{dp}{dx}=-30000.0$ Pa/m. So how to I account for the pressure gradient? Any help is SUPER appreciated. And no, this is not my research itself. This is something my professor gave me as a project in preparation for our research. I don't need an outright answer...just some guidance. Thanks!

H0T_S0UP · March 30, 2015, 15:46

Take a look at pressure-velocity coupling. 12 Steps to Navier-Stokes gives a good introduction and introduces this method, but when I looked at these lectures I felt they were seriously lacking depth.

Or...if you have a temperature boundary condition too and assume IC flow, you can use the equation of state P = pRT and solve the energy equation with momentum. This would make more sense physically, since you will lose energy to heat, but will introduce another level of complexity to your problem. You also need at least one boundary condition for your pressure.

Dale Andersons book is really thorough and probably gives insight into your problem, but it isn't for beginners.

Alex C. · March 30, 2015, 15:49

The most basic first derivative with central difference in space is :

$\frac{\partial \phi}{\partial x}\approx \frac{\phi^{n}_{j,k+1}-\phi^n_{j,k-1}}{2 \Delta t}$

This equation is the classic example of odd-even decoupling.

May I ask where is the advection term, or why it's not there?

DA6righthand · March 30, 2015, 16:21

The form of the NS equation I gave in the original post is what the book gave me. Actually, I'm using the Tannehil/Anderson book to learn from. The given problem is from an older edition of the book (I have the most recent edition).

As I said, I'm relatively new to CFD. I struggled through chapter 2 of the Anderson book (I taught myself basic PDE methods like separation of variables via my differential equations book) but my professor felt I had a good enough grasp on the material to move on to chapter 3. I'm currently in chapter 3 of Anderson which I find much easier than chapter 2 because of a numerical analysis course I previously took.

My first reaction to the problem in my first post was to let $\frac{\partial p}{\partial x}$ be constant but I'm completely unsure of that. But if I use the central difference, I'll end up with pressure and velocity terms mixed together which leads me to confusion when trying to use von Neumann analysis to determine the stability criteria. Also, I pretty much typed the problem from the book verbatim so everything about the problem is given above. Hopefully I didn't confuse you guys...

H0T_S0UP · March 30, 2015, 16:25

Quote:

The form of the NS equation I gave in the original post is what the book gave me. Actually, I'm using the Tannehil/Anderson book to learn from. The given problem is from an older edition of the book (I have the most recent edition).

As I said, I'm relatively new to CFD. I struggled through chapter 2 of the Anderson book (I taught myself basic PDE methods like separation of variables via my differential equations book). I'm currently in chapter 3 of Anderson which I find much easier than chapter 2 because of a numerical analysis course I previously took.

My first reaction to the problem in my first post was to let

be constant but I'm completely unsure of that. But if I use the central difference, I'll end up with pressure and velocity terms mixed together which leads me to confusion when trying to use von Neumann analysis to determine the stability criteria. Also, I pretty much typed the problem from the book verbatim so everything about the problem is given above. Hopefully I didn't confuse you guys...

Ha! I ran into a similar problem and asked my professor what to do and he flatly said "I don't know how that affects the stability analysis." That is a weakness of Andersons book, it is not accessibly to people just starting out. I kept it around, but have had lots of trial and error finding decent literature. NPTEL has good lectures on CFD that might help, but since that problem came so early in the book I doubt you need to do anything fancy like pressure-velocity coupling or adding the energy equation.

agd · March 30, 2015, 16:54

If you are imposing the pressure gradient then simply treat it as a source term. The problem description certainly makes it sound like that is how you should proceed.

DA6righthand · April 1, 2015, 18:33

So here's where I'm at. From what has been discussed thus far and from the problem statement, we can say pressure varies linearly in the x-direction and hence $\frac{dp}{dx}$ is a constant. Using the FTCS scheme yields

$\frac{u^{n+1}_{j,k}-u^n_{j,k}}{\Delta t}=\nu \frac{u^n_{j,k+1}-2u^n_{j,k}+u^n_{j,k-1}}{\Delta y^2}-\frac{1}{\rho }\frac{dp}{dx}$

And then solving for the

time is

$u^{n+1}_{j,k}=u^n_{j,k} +\frac{\nu \Delta t}{\Delta y^2}\left(u^n_{j,k+1}-2u^n_{j,k}+u^n_{j,k-1}\right)-\frac{\Delta t}{\rho }\frac{dp}{dx}$

Now I need to determine the stability criteria via von Neumann analysis. Because we only have an equation that varies in time and with height

, my first reaction is to let $u^n_{j,k}=e^{at}e^{ik_yy}$ . I know that after I substitute the exponentials into the discretized PDE, the amplification factor $G=e^{a\Delta t}$ and that $\left | G \right |\leq 1$ . I'm at a loss of what to do with the $\frac{\Delta t}{\rho }\frac{dp}{dx}$ term though after trying to find

. I end up with complex exponentials which depend on y and t in the equation for

which doesn't seem correct.

FMDenaro · April 2, 2015, 03:06

the last term is just a number...

a suggestion, the factor dt*mu/dy^2 = alpha, therefore the Von Neumann analysis should be performed checking the maximum of |G| for a range of values for alpha, dt and wavenumber.

FMDenaro · April 2, 2015, 04:14

have a look at:

http://www.google.it/url?sa=t&rct=j&...5ZeC4pGZuV7YwA

March 30, 2015, 14:57	Using FTCS to solve a reduced form of Navier-Stokes eqn	#1
DA6righthand New Member Justin Join Date: Jan 2015 Posts: 28 Rep Power: 11	I'm relatively new to CFD and took on research in CFD. I'm a mechanical engineering major...so I'll do my best in describing the given problem. The problem regards two parallel plates extended to infinity at a distance of h apart. The fluid between the plates has a given density and kinematic viscosity. The upper plate is stationary and the lower plate is suddenly set in motion with a constant velocity of 40 m/s. The spacing h is 4 cm. A constant streamwise pressure gradient of dp/dx is imposed within the domain at the instant motion starts. A spacial size of 0.001 m is specified. The reduced form of the Navier-Stokes equation is $\frac{\partial u}{\partial t}=\nu \frac{\partial ^2u}{\partial y^2}-\frac{1}{\rho }\frac{\partial p}{\partial x}$ Using finite differences, I get the following $\frac{\partial u}{\partial t}\approx \frac{u^{n+1}_{j,k}-u^n_{j,k}}{\Delta t}$ $\frac{\partial ^2u}{\partial y^2}\approx \frac{u^n_{j,k+1}-2u^n_{j,k}+u^n_{j,k-1}}{\Delta y^2}$ I'm confused about the $\frac{\partial p}{\partial x}$ term. I'm supposed to solve for velocity using the FTCS explicit scheme with 1) $\frac{dp}{dx}=0.0$ , 2) $\frac{dp}{dx}=20000.0$ Pa/m, and 3) $\frac{dp}{dx}=-30000.0$ Pa/m. So how to I account for the pressure gradient? Any help is SUPER appreciated. And no, this is not my research itself. This is something my professor gave me as a project in preparation for our research. I don't need an outright answer...just some guidance. Thanks! Last edited by DA6righthand; March 30, 2015 at 15:05. Reason: Accidentally posted before finishing post

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March 30, 2015, 15:46		#2
H0T_S0UP Member Alex Join Date: Jan 2014 Posts: 54 Rep Power: 12	Take a look at pressure-velocity coupling. 12 Steps to Navier-Stokes gives a good introduction and introduces this method, but when I looked at these lectures I felt they were seriously lacking depth. Or...if you have a temperature boundary condition too and assume IC flow, you can use the equation of state P = pRT and solve the energy equation with momentum. This would make more sense physically, since you will lose energy to heat, but will introduce another level of complexity to your problem. You also need at least one boundary condition for your pressure. Dale Andersons book is really thorough and probably gives insight into your problem, but it isn't for beginners.

March 30, 2015, 15:49		#3
Alex C. Member Join Date: Jul 2013 Posts: 55 Rep Power: 12	The most basic first derivative with central difference in space is : $\frac{\partial \phi}{\partial x}\approx \frac{\phi^{n}_{j,k+1}-\phi^n_{j,k-1}}{2 \Delta t}$ This equation is the classic example of odd-even decoupling. May I ask where is the advection term, or why it's not there?

March 30, 2015, 16:21		#4
DA6righthand New Member Justin Join Date: Jan 2015 Posts: 28 Rep Power: 11	The form of the NS equation I gave in the original post is what the book gave me. Actually, I'm using the Tannehil/Anderson book to learn from. The given problem is from an older edition of the book (I have the most recent edition). As I said, I'm relatively new to CFD. I struggled through chapter 2 of the Anderson book (I taught myself basic PDE methods like separation of variables via my differential equations book) but my professor felt I had a good enough grasp on the material to move on to chapter 3. I'm currently in chapter 3 of Anderson which I find much easier than chapter 2 because of a numerical analysis course I previously took. My first reaction to the problem in my first post was to let $\frac{\partial p}{\partial x}$ be constant but I'm completely unsure of that. But if I use the central difference, I'll end up with pressure and velocity terms mixed together which leads me to confusion when trying to use von Neumann analysis to determine the stability criteria. Also, I pretty much typed the problem from the book verbatim so everything about the problem is given above. Hopefully I didn't confuse you guys...

March 30, 2015, 16:54		#6
agd Senior Member Join Date: Jul 2009 Posts: 353 Rep Power: 18	If you are imposing the pressure gradient then simply treat it as a source term. The problem description certainly makes it sound like that is how you should proceed.

April 1, 2015, 18:33		#7
DA6righthand New Member Justin Join Date: Jan 2015 Posts: 28 Rep Power: 11	So here's where I'm at. From what has been discussed thus far and from the problem statement, we can say pressure varies linearly in the x-direction and hence $\frac{dp}{dx}$ is a constant. Using the FTCS scheme yields $\frac{u^{n+1}_{j,k}-u^n_{j,k}}{\Delta t}=\nu \frac{u^n_{j,k+1}-2u^n_{j,k}+u^n_{j,k-1}}{\Delta y^2}-\frac{1}{\rho }\frac{dp}{dx}$ And then solving for the time is $u^{n+1}_{j,k}=u^n_{j,k} +\frac{\nu \Delta t}{\Delta y^2}\left(u^n_{j,k+1}-2u^n_{j,k}+u^n_{j,k-1}\right)-\frac{\Delta t}{\rho }\frac{dp}{dx}$ Now I need to determine the stability criteria via von Neumann analysis. Because we only have an equation that varies in time and with height , my first reaction is to let $u^n_{j,k}=e^{at}e^{ik_yy}$ . I know that after I substitute the exponentials into the discretized PDE, the amplification factor $G=e^{a\Delta t}$ and that $\left \| G \right \|\leq 1$ . I'm at a loss of what to do with the $\frac{\Delta t}{\rho }\frac{dp}{dx}$ term though after trying to find . I end up with complex exponentials which depend on y and t in the equation for which doesn't seem correct.

April 2, 2015, 03:06		#8
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,768 Rep Power: 71	the last term is just a number... a suggestion, the factor dt*mu/dy^2 = alpha, therefore the Von Neumann analysis should be performed checking the maximum of \|G\| for a range of values for alpha, dt and wavenumber.

April 2, 2015, 04:14		#9
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,768 Rep Power: 71	have a look at: http://www.google.it/url?sa=t&rct=j&...5ZeC4pGZuV7YwA