Discretization of the convective term of the N-S equation

Donton · August 31, 2017, 06:06

Hello everyone...

In most of the document explaining the Fractional Step Method I have found that the non-dimansionl for of the Navier-Stokes equation is discretized as,

(u^n+1 - u^n)/dt + (3/2)*H(u^n) - (1/2)*H(u^n-1) = ... the right hand side

And it is mentioned the H is the discrete convective operator. Now my question is how to expand this H?

I have done in the following way. Please tell me if I am incorrect.
Considering the non-dimentional form of the x-momentum equation

H(u^n) = ( (u^n * u^n)|i+1/2,j - (u^n * u^n)|i-1/2,j )/dx + ( (v^n * u^n)|i,j+1/2 - (v^n * u^n)|i,j-1/2 )/dy

here ^n denotes the superscript n for time level n
Now if we write the equation without the superscript just for simlicity,

H(u^n) = ( (u*u)|i+1/2,j - (u*u)|i-1/2,j )/dx + ( (v*u)|i,j+1/2 - (v*u)|i,j-1/2 )/dy

Now considering uniform structurred staggered grid arrangement and central difference scheme,

(u*u)|i+1/2,j = u|i+1/2,j * u|i+1/2,j
= ((u|i+1,j + u|i,j)/2) * ((u|i+1,j + u|i,j)/2)

(u*u)|i-1/2,j = u|i-1/2,j * u|i-1/2,j
= ((u|i-1,j + u|i,j)/2) * ((u|i-1,j + u|i,j)/2)

(v*u)|i,j+1/2 = v|i,j+1/2 * u|i,j+1/2
= ((v|i+1,j + v|i,j)/2) * ((u|i,j+1 + u|i,j)/2)

(v*u)|i,j-1/2 = v|i,j-1/2 * u|i,j-1/2
= ((v|i+1,j-1 + v|i,j-1)/2) * ((u|i,j-1 + u|i,j)/2)

Is this discretization correct? My next question is if I want to use upwind scheme how will it be?

Also in Fractional step method the convective term is estimated based on the previous time step's values. Therefore while solving velocity at each point from the momentum equation for the current time step the convective term will just become a known quantity and will go to the right hand side of the algebriac equations. Then how does it effect the solution wheather central difference or upwinding is used for the discretization of the convective term?

FMDenaro · September 1, 2017, 12:01

Well, what you see is the explicit multi-step Adams-Bashforth second order accurate time integration applied only for the convective term vector H=Div(vv). This notation is used in the hystorical work of Kim and Moin, (for example see https://ntrs.nasa.gov/archive/nasa/c...9840014260.pdf)
Then, you can introduce any type of spatial discretization you want to H^n and H^n-1. Of course, different final algebric terms appear if you use staggered or non-staggered grids, central or upwind schemes of various order. Any consistent discretization is correct, there is no a unique possible discretization. And that does not enter into the Fractional Step Method.

August 31, 2017, 06:06	Discretization of the convective term of the N-S equation	#1
Donton New Member Donton Join Date: Aug 2017 Posts: 6 Rep Power: 8	Hello everyone... In most of the document explaining the Fractional Step Method I have found that the non-dimansionl for of the Navier-Stokes equation is discretized as, (u^n+1 - u^n)/dt + (3/2)H(u^n) - (1/2)H(u^n-1) = ... the right hand side And it is mentioned the H is the discrete convective operator. Now my question is how to expand this H? I have done in the following way. Please tell me if I am incorrect. Considering the non-dimentional form of the x-momentum equation H(u^n) = ( (u^n * u^n)\|i+1/2,j - (u^n * u^n)\|i-1/2,j )/dx + ( (v^n * u^n)\|i,j+1/2 - (v^n * u^n)\|i,j-1/2 )/dy here ^n denotes the superscript n for time level n Now if we write the equation without the superscript just for simlicity, H(u^n) = ( (uu)\|i+1/2,j - (uu)\|i-1/2,j )/dx + ( (vu)\|i,j+1/2 - (vu)\|i,j-1/2 )/dy Now considering uniform structurred staggered grid arrangement and central difference scheme, (uu)\|i+1/2,j = u\|i+1/2,j u\|i+1/2,j = ((u\|i+1,j + u\|i,j)/2) * ((u\|i+1,j + u\|i,j)/2) (uu)\|i-1/2,j = u\|i-1/2,j u\|i-1/2,j = ((u\|i-1,j + u\|i,j)/2) * ((u\|i-1,j + u\|i,j)/2) (vu)\|i,j+1/2 = v\|i,j+1/2 u\|i,j+1/2 = ((v\|i+1,j + v\|i,j)/2) * ((u\|i,j+1 + u\|i,j)/2) (vu)\|i,j-1/2 = v\|i,j-1/2 u\|i,j-1/2 = ((v\|i+1,j-1 + v\|i,j-1)/2) * ((u\|i,j-1 + u\|i,j)/2) Is this discretization correct? My next question is if I want to use upwind scheme how will it be? Also in Fractional step method the convective term is estimated based on the previous time step's values. Therefore while solving velocity at each point from the momentum equation for the current time step the convective term will just become a known quantity and will go to the right hand side of the algebriac equations. Then how does it effect the solution wheather central difference or upwinding is used for the discretization of the convective term?

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September 1, 2017, 12:01		#2
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,768 Rep Power: 71	Well, what you see is the explicit multi-step Adams-Bashforth second order accurate time integration applied only for the convective term vector H=Div(vv). This notation is used in the hystorical work of Kim and Moin, (for example see https://ntrs.nasa.gov/archive/nasa/c...9840014260.pdf) Then, you can introduce any type of spatial discretization you want to H^n and H^n-1. Of course, different final algebric terms appear if you use staggered or non-staggered grids, central or upwind schemes of various order. Any consistent discretization is correct, there is no a unique possible discretization. And that does not enter into the Fractional Step Method.