By R.D. Richtmyer

ISBN-10: 3540071393

ISBN-13: 9783540071396

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Thus, only continuum models can be used for macroscopic phenomena of ﬂuid ﬂow. 4 Incompressible Flow In an incompressible ﬂuid the density ρ does not change if we follow the motion of the ﬂuid particles of the ﬂow. We can express this fact in the diﬀerential equation form Du ρ ≡ ρ˙ + u · ∇ρ = 0, where Du ρ is the convective derivative of ρ with respect to the velocity u. We obtain the convective derivative by computing the change in time following the trajectory x(t) of a ﬂuid particle satisfying the diﬀerential equation x(t) ˙ = u(x, t).

20 1 Main Objective Fig. 16. From Chapter 35: Adaptive mesh reﬁnement for the ﬂow past a NACA 0012: magnitude of the velocity (upper), dual solution (middle) representing sensitivity information related to the computation of lift and drag, and a corresponding (coarse) mesh under reﬁnement (lower). 1 Computational Turbulent Incompressible Flow 21 Fig. 17. From Chapter 35: Midsections showing snapshots of a G2 simulation of the blood ﬂow in a realistic bifurcation model of a human carotid bifurcation (upper), the dual solution corresponding to the computational error in wall shear stress (middle), and the corresponding mesh (lower).

In Sommerfeld’s Mystery from 1908 [104], mathematics predicts that the simplest of all ﬂows, Couette ﬂow with a stationary linear velocity proﬁle, is stable and thus should exist. But nobody has ever observed this ﬂow in a ﬂuid with small viscosity. The cover up of d’Alembert’s Mystery is to blame the assumption of zero viscosity for the erroneous prediction: In reality there is always some possibly very very small viscosity (of some nature), which changes everything. We will below argue that such explanations are not scientiﬁcally satisfactory and we shall instead present a new resolution based on computational turbulence in the inviscid Euler equations.

### 4th Int'l Conference on Numerical Methods in Fluid Dynamics by R.D. Richtmyer

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