It is common to divide these equations by ρ since it is constant and express

It is common to divide these equations by ρ since it

This preview shows page 85 - 87 out of 164 pages.

It is common to divide these equations by ρ (since it is constant), and express them as u t + uu x + vu y + wu z = 1 ρ p x + ν u + 1 ρ F B,x , (3.43a) v t + uv x + vv y + wv z = 1 ρ p y + ν v + 1 ρ F B,y , (3.43b) w t + uw x + vw y + ww z = 1 ρ p z + ν w + 1 ρ F B,z . (3.43c) Here, ν is kinematic viscosity , the ratio of viscosity μ to density ρ , as given earlier in Chap. 2, and ∆ is the second-order partial differential operator (given here in Cartesian coordinates) ∆ = 2 ∂x 2 + 2 ∂y 2 + 2 ∂z 2 known as the Laplacian or Laplace operator (which is usually denoted by 2 in the engineering and physics literature). The notation shown here is modern and is becoming increasingly widespread. These equations are called the Navier–Stokes equations , and they provide a pointwise descrip- tion of essentially any time-dependent incompressible fluid flow. But it is important to recall the assumptions under which they have been derived from Newton’s second law of motion applied to a fluid element. First, the continuum hypothesis has been used repeatedly to permit pointwise defini- tions of various flow properties and to allow definition of fluid elements. Second, we have assumed constant density ρ , and corresponding to this a divergence-free velocity field; i.e. , ∇ · U = 0. In addition, we have invoked (a generalization of) Newton’s law of viscosity to provide a formulation for shear stresses, and tacitly assumed an analogous result could be used to define normal viscous
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80 CHAPTER 3. THE EQUATIONS OF FLUID MOTION stresses; furthermore, we have taken viscosity to be constant. Finally, from a mathematical per- spective, we are implicitly assuming velocity and pressure fields are sufficiently smooth to permit all indicated differentiations. While the composite of these assumptions may seem quite restrictive, in fact they are satisfied by many physical flows, and we will not in these lectures dwell much on cases in which they may fail. But we note that at least for relatively low-speed flows all of the above assumptions essentially always hold to a good approximation. As the flow speed increases, constant density (and, consequently, also the divergence-free condition) fails as we move into compressible flow regimes, but in general this occurs only for gaseous flows. Constant viscosity is an extremely good assumption provided temperature is nearly constant, and for flows of gases it is generally a good approximation until flows become compressible. The mathematical smoothness assumption is possibly the most likely to be violated, but from an engineering perspective this is usually not a consideration. 3.5 Analysis of the Navier–Stokes Equations In this section we will first provide a brief introduction to the mathematical structure of Eqs. (3.43) which is particularly important when employing CFD codes to solve problems in fluid dynamics.
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