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CHAPTER 2. PRELIMINARIES 34
Remark 2.18. Two possible problems of this preconditioner can arise of the assumption
(2.42). First, the commutativity
∇[(w · ∇)] ≈ (w · ∇)∇,
is not fulfilled in general (except for some special situation, e.g. constant w), what poses
problems if this term dominates (for small ν’s).
Second, for non-constant ν (e.g. due to a k-ɛ turbulence model, c.f. [MP94] or [RW99])
even the first part of (2.42) would be violated, as in this case
∇ν∆ ≠ ν∆∇.