Use of Relaxation Factors in CFD Analysis
Use of Relaxation Factors in CFD Analysis
Field and Equation Under-relaxations:
Under-relaxation is a fraction, often used in iterative methods to stabilize the linear solvers. Let’s assume a transport equation is solved for variable ϕ. As iterative method is being followed, the value of ϕ can get updated in subsequent iterations. At beginning of present iteration let’s denote the transport variable as ϕpi and after the iteration the variable value as ϕai. The difference (ϕai − ϕpi) can be under-relaxed by a factor α and added back to ϕpi for the next iteration. Hence for the next iteration the value of ϕ would be ϕpi + α(ϕai − ϕpi). This kind of under-relaxation is applied after evaluating the value of ϕ after each iteration and is known as field under-relaxation. As it requires to store a new variable this kind of under-relaxation require more computational resource.
However, instead of storing an additional variable, the concept of under-relaxation can be explicitly included in transport equations, then this is known as equation under-relaxation.
Diagonal Dominance of Matrix and Under-relaxation:
The iterative methods have specific known sufficient criteria for convergence. For example, the sufficient criteria for convergence for Gauss Seidel is a diagonal-dominated matrix (weak diagonal dominance). Normally transport equations for fluid flow satisfy weak-diagonal dominant condition. However, it is not always guaranteed to have a diagonal dominant matrix due to advection terms present in transport equations. A weak diagonal dominant matrix has diagonal terms larger or equal in magnitude than sum of magnitude of other entries in a row. To read in more details about diagonal dominance section 5.5 of the book by Greenshields and Weller “Notes on Computational Fluid Dynamics: General Principles” can be referred. Dividing the diagonal terms with fractions can help making the magnitude of diagonal terms larger. This is how the under-relaxations can help.
Pressure velocity coupling algorithms and under-relaxation:
The requirement of under-relaxation is also linked to the pressure and velocity coupling algorithm chosen to solve a transport equation. SIMPLE algorithm is based on an assumed pressure field. Also, during the algorithmic steps, it ignores specific terms, which introduces an error in the iteration. Therefore, an under-relaxation factor is required for pressure and velocity fields. In contrast, SIMPLEC algorithm does not have this limitation, hence an under-relaxation for pressure is not required. However, an under-relaxation in velocity and other scalars are provided in both SIMPLE and SIMPLEC. The PISO algorithm, being used for transient simulations, does not require any under-relaxation as the transient equations with Co < 1, would be diagonally dominant due to higher magnitude of diagonal term (related to time derivatives) over the advection terms.
Value ranges of under-relaxation:
The lower the under-relaxation, more stable is the solution. However, a value around 0.2 is highly restrictive for the solution to march. Such a lower relaxation indicates a problem in combination of schemes and boundary conditions. Hence, suitable change should be made in the CFD solver settings. A higher relaxation ( ≥ 0.7) is always desirable. Also, apart from stability of solver, the residual plots should be monitored and the relaxation should be altered to achieve a lower order residual.
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