Notes on numerical fluid mechanics, vol.14: Finite approximations in fluid mechanics

Fluid Mechanics : Numerical Methods

where σ is the stress tensor, ǫ = 12 (∇u +∇u) T is the strain tensor, f is a body force per unit mass (gravity is a typical example), qT is a volume source (it may model chemical reactions, Joule effects, radioactive decay, etc.), and jT is the heat flux. In addition to the above three fundamental conservation equations, one may also have to add L equations that accounts for the conservation of...

Lecture Notes on Intermediate Fluid Mechanics

Notes #12 MAE 533, Fluid Mechanics

We now consider the classical theory of thin airfoils. The generic problem is: we have a uniform, steady low speed (M 2 ∞ << 1) flow with undisturbed velocity U∞ in the +x direction over a thin wing with known planform and wing cross-sections located in the vicinity of the y ≈ 0 plane. The plan form is specified by the “chord distribution,” c(z). The “root chord” of the wing is denoted by co = ...

Notes #6 MAE 533, Fluid Mechanics

1 Different Ways of Representing T • The speed of sound, a, is formally defined as (∂p/∂ρ) s. It is a function of that thermodynamic state variables of the gas. The stagnation enthalphy of a steady flow streamtube is defined by " h • ≡ h + 1 2 u 2. (1) It depends not only of the thermodynamic state of the gas, but also on the magnitude of the flow velocity u. Using the perfect gas approximation...

Notes #8 MAE 533, Fluid Mechanics

1 One-dimensional Unsteady Inviscid Flows We consider now the problem of unsteady flow in one-dimension under the inviscid assumption. We have a rigid tube with cross-sectional area A(x), and we assume that all the variables (x-component of velocity u and all the thermodynamic state variables) are functions of x and t only. The governing equations are: A Dρ Dt + ρ ∂(uA) ∂x = 0, (1a) ρ Du Dt + ∂...