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Additional info for An introduction to computational fluid dynamics
These lead to severe difﬁculties in the speciﬁcation of boundary conditions for completely generalpurpose CFD procedures working at any Reynolds number and Mach number. Experience with the ﬁnite volume method has yielded a set of auxiliary conditions that give physically realistic ﬂow solutions in many industrially relevant problems. qxd 29/12/2006 04:34PM Page 40 Chapter three Turbulence and its modelling All ﬂows encountered in engineering practice, simple ones, such as twodimensional jets, wakes, pipe ﬂows and ﬂat plate boundary layers, and more complicated three-dimensional ones, become unstable above a certain Reynolds number (UL/ν where U and L are characteristic velocity and length scales of the mean ﬂow and ν is the kinematic viscosity).
A chaotic and random state of motion develops in which the velocity and pressure change continuously with time within substantial regions of ﬂow. Flows in the laminar regime are completely described by the equations developed in Chapter 2. In simple cases the continuity and Navier–Stokes equations can be solved analytically (Schlichting, 1979). More complex ﬂows can be tackled numerically with CFD techniques such as the ﬁnite volume method without additional approximations. Many, if not most, ﬂows of engineering signiﬁcance are turbulent, so the turbulent ﬂow regime is not just of theoretical interest.
55) We look at the behaviour within the region −1 < y < 1. Hence a = a(x, y) = y, b = 0 and c = 1. The value of discriminant (b2 − 4ac) is equal to −4y. 55) is of mixed type. The equation is locally hyperbolic, parabolic or elliptic depending on the value of y. For the non-linear case similar remarks apply. The classiﬁcation of the PDE depends on the local values of a, b and c. Second-order PDEs in N independent variables (x1, x2, . . 56) k Fletcher (1991) explains that the equation can be classiﬁed on the basis of the eigenvalues of a matrix with entries Ajk.