<< Chapter < Page Chapter >> Page >

This is the equation when the coordinates have the same reference length. If the coordinates are normalized with respect to different lengths, then the equation is as follows.

2 P x 2 + α 2 2 P y 2 = 2 α 2 2 ψ x 2 2 ψ y 2 - 2 ψ x y 2

The Poisson equation for pressure needs to have boundary conditions for solution. The equations of motion give an expression for the pressure gradient. The normal derivative of pressure at the boundary can be determined for the Neumann-type boundary condition. The equations of motion usually simplify at boundaries if the boundary has no slip BC or if it has parallel flow in the direction either parallel or normal to the boundary. For example, for no-slip

P x = α 2 R e 2 u y 2 , at y = c , u = v = 0 α P y = 1 R e 2 v x 2 , at x = c , u = v = 0

If the flow is parallel and in the direction normal to the boundary

P x = α 2 R e 2 u y 2 , y = c , v = 0 , u = u ( y ) α P y = 1 R e 2 v x 2 , x = c , u = 0 , v = v ( x )

Cylindrical-polar coordinates

The code for the numerical solution of the Navier-Stokes equations in Cartesian coordinates can be easily modified to cylindrical-polar coordinates. The coordinate transformation is first illustrated for the Laplacian operator.

2 ψ = 1 r r r ψ r + 1 r 2 2 ψ θ 2 , r 1 r r 2 , 0 θ θ o

The independent variables are made dimensionless with respect to the boundary parameters.

r * = r r 1 , θ * = θ θ o , 1 r * r 2 r 1 = β , 0 θ * 1

The Lapacian operator with the dimensionless coordinates after dropping the * is now,

2 ψ = 1 r 1 2 1 r r r ψ r + 1 θ o 2 1 r 2 2 ψ θ 2 , 1 r β , 0 θ 1

The radial coordinate is transformed to the logarithm of the radius.

z = ln r ln β = γ ln r , 0 z 1 , γ = 1 ln r 2 / r 1 r = exp z / γ r = d z d r z = γ r z 1 r r r ψ r = γ 2 r 2 2 ψ z 2

The Laplacian operator is now as follows,

2 ψ = 1 r 1 2 γ 2 r 2 2 ψ z 2 + α 2 r 2 2 ψ θ 2 , 0 z 1 , 0 θ 1 , α = 1 θ o

The finite difference expression for the Laplacian will be same as that for Cartesian coordinates with ( z , θ ) substituted for ( x , y ) except for γ 2 and r 2 factors.

0 z i 1 , 0 θ j 1 , i , j = 1 , 2 , ... , J M A X δ = 1 J M A X - 1 z i = δ i - 1 , θ j = δ j - 1

The curl operator is modified from that in Cartesian coordinates.

v r = α r ψ θ v θ = - ψ r = - γ r ψ z w = 1 r r v θ r - α r v r θ = γ r 2 r v θ z - α r v r θ = - γ 2 r 2 2 ψ z 2 - α 2 r 2 2 ψ θ 2

The vorticity boundary conditions with the transformed coordinates is for the z boundary,

w 1 B C = - γ 2 r 2 2 δ 2 ψ 2 - ψ 1 B C γ r 2 δ v θ B C - α r v r B C θ

and for the θ boundary,

w 1 B C = - α 2 r 2 2 δ 2 ψ 2 - ψ 1 B C ± α r 2 δ v r B C + γ r 2 r v θ B C z

The stream function at the boundaries are expressed different from that in Cartesian coordinates.

d ψ = r v r / α d θ , at z boundary d ψ = - r v θ / γ d z , at θ boundary

The convective terms are expressed different from that in Cartesian coordinates.

u w x 1 r r u w r = γ r 2 r u w z γ r 2 δ r u w i + 1 / 2 - r u w i - 1 / 2 α v w y α r v w θ α r δ v w j + 1 / 2 - v w j - 1 / 2

The code should be written so one can have either Cartesian coordinates or transformed cylindrical-polar coordinates. A parameter will be needed to identify the choice of coordinates, e.g. i c a s e = 1 for Cartesian coordinates and i c a s e = 2 for transformed cylindrical-polar coordinates. Also, another parameter should be specified to identify the choice of boundary conditions, e.g., i b c = 1 for radial flow, i b c = 2 for Couette flow, and i b c = 3 for flow around a cylinder. Test cases with known solutions should be used to verify the code. The first case is radial, potential flow from a line source and the second is Couette flow in the annular region between two cylindrical surfaces.

v r = 1 / r v θ = 0 radial flow v r = 0 v θ = r - 1 / r / β - 1 / β Couette flow

Flow around a cylinder needs a boundary condition for the flow far away from the cylinder. The flow very far away may be uniform translation. However, this condition may be so far away that it may result in loss of resolution near the cylinder. Another boundary condition that may be specified beyond the region of influence of the boundary layer is to use the potential flow past a cylinder. This boundary condition will not be correct in the wake of cylinder where the flow is disturbed by the convected boundary layer buts its influence may be minimized if the outer boundary is far enough.

v r = 1 - 1 / r 2 cos θ v θ = - 1 + 1 / r 2 sin θ Potential flow past cylinder

Potential flow will not be a valid approximation along the θ boundaries close to the cylinder. Here, one may assume a surface of symmetry, at least for the upstream side.

Questions & Answers

what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Transport phenomena. OpenStax CNX. May 24, 2010 Download for free at http://cnx.org/content/col11205/1.1
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Transport phenomena' conversation and receive update notifications?

Ask