Calculation of The Flow Field

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Calculation of The Flow Field

• Patrick Mensah, PhD

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Introduction

• Introduction of need in calculating flow field
  variable u, v, w, P
• Difficulties in calculating flow field variable
  u, v, w, P
• Representation of the Pressure Gradient Term
• Representation of the Continuity Equation
• Recommended Remedy Staggered Grid Approach

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Introduction

• The Momentum Equations
• The Pressure and Velocity Corrections
• The Pressure-Corrections Equation
• The SIMPLE Algorithm
• A Revised Algorithm SIMPLER
• Summary

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Introduction of need in calculating flow field
variables

• Low field variable are usually unknown
• Local flow field velocity components are usually
  calculated from governing equations
• Even though flow field velocities can be obtained
  from the momentum equation, the difficulty is
  that the pressure gradient term is unknown

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Introduction of need in calculating flow field
variables

• Some approaches to resolving unknown pressure
  gradient issue
• Use of Vorticity-Based Methods This approach
  eliminates the pressure from the governing
  equations- do problem 6.1
• Results in the solution of two equation for the
  stream function and the vorticity
• Disadvantage is that
• it is not applicable in 3-D flows since stream
  functions do not exist,
• pressure may be needed in calculating flow
  properties such as density

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Difficulties in calculating flow field variable
u, v, w, P

• Representation of the Pressure Gradient Term
• Integrating over the control volume results in
  Pw-Pe contribution to the discretization equation
  

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Difficulties in calculating flow field variable
u, v, w, P

• This gives us the net pressure force exerted on
  the control volume
• In order to represent the net pressure in term s
  of grid point values a piecewise linear profile
  is assume giving
• The result is that the momentum equation will
  contain pressure difference between two alternate
  grid points and not adjacent ones

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Difficulties in calculating flow field variable
u, v, w, P

• Gives unrealistic pressure field

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Difficulties in calculating flow field variable
u, v, w, P

• Representation of the Continuity Equation
• Again results in unrealistic velocity field which
  demands equality of velocities at alternate grid
  points

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Recommended Remedy Staggered Grid Approach

• Recognizing that calculation of all variables
  does not have to be at the same grid point hence
  location of u velocities

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Recommended Remedy Staggered Grid Approach

• location of u and v velocities

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Recommended Remedy Staggered Grid Approach

• Advantages
• Discretized continuity equation would contain
  differences of adjacent velocity components
• The staggered grid results in a pressure
  difference between adjacent points that becomes
  the driving force for the velocity components
  located between the grid points

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The Momentum Equations

• Control volume for u

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The Momentum Equations

• Control volume for v

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The Momentum Equations

• When the pressure field is given or estimated the
  velocity field can be obtained from the following
  discretized equations momentum equations
• Where , u, v and w denotes a starred or
  estimated velocity field from P

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The Pressure and Velocity Corrections
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The Pressure Correction Equation

• Continuity Equation

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The Pressure Correction Equation
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The Pressure Correction Equation
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SemiImplicit Method for Pressure-Linked
Equations (SIMPLE ALGORITHM)

• Guess the pressure field P.
• Solve the momentum equations, such as Eqs.
  (6.8)-(6.10), to obtain u, v w
• Solve the p' equation.
• Calculate p from Eq. (6.11) by adding P' to P.
• Calculate u, v, w from their starred values using
  the velocity-correction formulas (6.17)-(6.19).

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SIMPLE ALGORITHM

• 6. Solve the discretization equation for other ?
  (such as temperature, concentration, and
  turbulence quantities) if they influence the flow
  field through fluid properties, source terms,
  etc. (If a particular ? does not influence the
  flow field, it is better to calculate it after a
  converged solution for the flow field has been
  obtained.)
• 7. Treat the corrected pressure p as a new
  guessed pressure p, return to step 2, and repeat
  the whole procedure until a converged solution is
  obtained.

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Things to note about the pressure correction
equation
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A Revised Algorithm SIMPLER

• Improves on the convergence of SIMPLE
• Improves obtaining corrected pressure field
• Faster convergence for 1-D problems
• No guess pressure required but extracts pressure
  from a given velocity field
• No need for further iterations when given
  velocities happen to be the correct velocity
  field
• However, more computational effort is required
  with SIMPLER than SIMPLE even though fewer
  iterations for convergence

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A Revised Algorithm SIMPLER

• Guess the velocity field v.
• Calculate the coefficients for the momentum
  equations and hence calculate from
  equations such as Eq. 6.26 by substituting the
  values of the neighbor velocities unb
• Calculate the coefficients for the pressure
  equation 6.30, and solve it to obtain the
  pressure field
• Treating the pressure field as P, solve the
  momentum equations, such as Eqs. (6.8)-(6.10), to
  obtain u, v w

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A Revised Algorithm SIMPLER

• 5. Calculate the mass source b eq. 6.23h and
  hence solve the p' equation.
• 6. Correct the velocity field u, v, w from their
  starred values using the velocity-correction
  formulas (6.17)-(6.19). Do not correct the
  pressure
• 7. Solve the discretization equation for other ?
  if necessary
• 8. Return to step 2, and repeat the whole
  procedure until a converged solution is obtained.