Advanced — Fluid Mechanics Problems And Solutions __link__
Equating the general stream function to this constant gives the profile equation:
Cfx=2(0.332)Rex=0.664Rexcap C sub f x end-sub equals the fraction with numerator 2 open paren 0.332 close paren and denominator the square root of Re sub x end-root end-fraction equals the fraction with numerator 0.664 and denominator the square root of Re sub x end-root end-fraction Final Answer The Blasius equation is with boundary conditions . The local skin friction coefficient is . 3. Compressible Flow: Oblique Shock Wave Relations Problem Statement A supersonic airflow at Mach , static pressure , and static temperature encounters a compression corner with a deflection angle of Assuming an ideal gas with , determine: The weak oblique shock wave angle ( The downstream Mach number ( M2cap M sub 2 The downstream static pressure ( Step-by-Step Solution Step 1: Find the Shock Angle ( ) using the The analytical relationship connecting deflection angle , shock angle , and upstream Mach number
The governing equations for a normal shock wave in an ideal gas yield the direct Prandtl relation for Mach numbers across a shock:
The velocity fields are derived above, and the total drag force equals (Stokes' Drag Law). 2. Boundary Layer Theory: Blasius Similarity Solution Problem Statement An incompressible fluid flows at velocity U∞cap U sub infinity end-sub advanced fluid mechanics problems and solutions
τw=μ𝜕u𝜕y|y=0=μU∞U∞νxf′′(0)tau sub w equals mu partial u over partial y end-fraction vertical line sub y equals 0 end-sub equals mu cap U sub infinity end-sub the square root of the fraction with numerator cap U sub infinity end-sub and denominator nu x end-fraction end-root f double prime of 0 The local skin friction coefficient Cfxcap C sub f x end-sub
𝜕u𝜕y=U∞U∞νxf′′(η)partial u over partial y end-fraction equals cap U sub infinity end-sub the square root of the fraction with numerator cap U sub infinity end-sub and denominator nu x end-fraction end-root f double prime of open paren eta close paren
Substitute $C_1$ and $C_2$ back into the equation: $$ u(y) = \fracU yB - \frac12\mu \left(-\fracdPdx\right) (By - y^2) $$ (Here, we typically define a favorable pressure gradient as negative, so we swap signs for clarity). Equating the general stream function to this constant
ψuniform=U∞rsinθ,ϕuniform=U∞rcosθpsi sub uniform end-sub equals cap U sub infinity end-sub r sine theta comma space phi sub uniform end-sub equals cap U sub infinity end-sub r cosine theta A line source at the origin emitting a volume flow rate per unit length has:
p(θ)=p∞+12ρU∞2−12ρ(-2U∞sinθ−Γ2πR)2p open paren theta close paren equals p sub infinity end-sub plus one-half rho cap U sub infinity end-sub squared minus one-half rho open paren negative 2 cap U sub infinity end-sub sine theta minus the fraction with numerator cap gamma and denominator 2 pi cap R end-fraction close paren squared Expanding the squared term:
η=yU∞νxeta equals y the square root of the fraction with numerator cap U sub infinity end-sub and denominator nu x end-fraction end-root the velocity must be finite
0=−G2μ(0)2+C1(0)+C2⟹C2=00 equals negative the fraction with numerator cap G and denominator 2 mu end-fraction open paren 0 close paren squared plus cap C sub 1 open paren 0 close paren plus cap C sub 2 ⟹ cap C sub 2 equals 0 At the top plate (
), the inertial terms in the Navier-Stokes equations can be neglected (Stokes flow / creeping flow).
The flow is a superposition of a linear velocity profile (Couette flow) and a parabolic profile (Poiseuille flow). 2. Potential Flow Theory & Superposition
vx(r)=r24μ(dpdx)+C1ln(r)+C2v sub x open paren r close paren equals the fraction with numerator r squared and denominator 4 mu end-fraction open paren d p over d x end-fraction close paren plus cap C sub 1 l n r plus cap C sub 2 Apply boundary conditions: At , the velocity must be finite, so . No-slip: At , . This gives