Transport equations for each velocity component – momentum equations – can be derived from the general transport equation by replacing the variable φ with u, v and w respectively.
The above equations govern a two-dimensional laminar steady ﬂow.
The Main Problem!
The velocity ﬁeld obtained from the momentum equation must also satisfy the continuity equation. The convective terms of the momentum equations contain non-linear quantities: for example, the ﬁrst term of the equation is the x derivative of ρu2.
All three equations are intricately coupled because every velocity component appears in each momentum equation and in the continuity equation.
The most complex issue to resolve is the role played by the pressure. It appears in both momentum equations, but there is evidently no (transport or other) equation for the pressure.
If the pressure gradient is known, the process of obtaining discretized equations for velocities from the momentum equations is exactly the same as that for any other scalar.
If the ﬂow is compressible the continuity equation may be used as the transport equation for density and, the energy equation is the transport equation for temperature. The pressure may then be obtained from density and temperature by using the equation of state p = p(ρ, T).
What happens when the flow is incompressible?
Issue with Incompressible Flow!
If the ﬂow is incompressible the density is constant and hence by deﬁnition not linked to the pressure. In this case coupling between pressure and velocity introduces a constraint in the solution of the ﬂow ﬁeld. We don’t have any separate equation for pressure.
Still, we need to supply the correct pressure field into the momentum equation so that the resulting velocity field satisfies the continuity constraint.
The pressure–velocity linkage can be resolved by adopting an iterative solution strategy such as the SIMPLE algorithm of Patankar and Spalding (1972).
In SIMPLE Algorithm:-
Ø The convective ﬂuxes through cell faces are evaluated from guessed velocity components.
Ø A guessed pressure ﬁeld is used to solve the momentum equations.
Ø A pressure correction equation is derived from the continuity & momentum equation.
Ø That Pressure Correction equation is solved to obtain a pressure correction ﬁeld, which is in turn used to update the velocity and pressure ﬁelds that will satisfy the continuity equation.
Ø Again Momentum equation is solved with the updated pressure and velocity field.
Ø The process is iterated until convergence of the velocity and pressure ﬁelds.
Principle behind SIMPLE Algorithm
Based on the premise that fluid flows from regions with high pressure to low pressure.
Detailed Algorithm Derivation
The SIMPLE stands for Semi-Implicit Method for Pressure-Linked Equations.
The algorithm is essentially a guess-and-correct procedure for the calculation of pressure on the staggered grid arrangement.
Now, the steps involved in solving the 2D incompressible Navier Stokes Equation using SIMPLE Algorithm are explained.
1. Guess & Solve
Guess a pressure ﬁeld, p* is guessed.
Discretised momentum equations are solved using the guessed pressure ﬁeld to yield velocity components u* and v* as follows:
u* & v* satisfies the momentum equation but does not satisfy the continuity equation.
The equation shown above is obtained from Finite Volume Discretization of Momentum Equation.
2. Introduce a Correction Term
We deﬁne the correction p′ as the difference between correct pressure ﬁeld p and the guessed pressure ﬁeld p*.
Similarly, we deﬁne velocity corrections u′ and v′ to relate the correct velocities u & v to the guessed velocities u* and v*.
3. Obtain Velocity Correction
4. Obtain an Equation for Pressure Correction P’
As mentioned above, Momentum Equation solved using guessed pressure field satisfies the momentum conservation but not the mass conservation.
Then we introduce a correction term P’, U’, V’ which is added with guessed value P*, U*, V* to correct the pressure and velocity field.
Then we obtain an expression that corrects the velocity field based on the guessed velocities U*, V* and corrected pressure P’ as shown below.
Now we have to obtain an equation for pressure correction term P’ so that we can correct our velocity field.
5. Correct the Pressure Field
6. Correct the Velocity Field