ME 521 Fall 2007 Professor John M. Cimbala Lecture 29 11/02/2007
Today, we will:
•
Discuss propulsion of fish, birds, and sailboats
•
Do Candy Questions for Candy Friday
•
Begin a new major topic: Laminar Flow Solutions
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Laminar Flow Solutions - Foundations of Fluid Mechanics I - Handout, Exercises of Fluid Mechanics
This is the first course of a two-semester fluid mechanics sequence for graduate students in the thermal sciences. This course deals with solutions of these equations, both exact and approximate. Key points of this lecture are: Laminar Flow Solutions, Propulsion of Fish, Birds, and Sailboats, Nearly Incompressible Laminar Flow, Equations and Solution Technique, Transitional or Turbulent, Conservation of Mass, Momentum Equation, Conservation of Energy, Vorticity Equation, Continuity and Vorticit
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ME 521 Fall 2007 Professor John M. Cimbala Lecture 29 11/02/
Today, we will :
• Discuss propulsion of fish, birds, and sailboats
• Do Candy Questions for Candy Friday
• Begin a new major topic: Laminar Flow Solutions
Nearly Incompressible Laminar Flow – Equations and Solution Technique
Author: John M. Cimbala, Penn State University Latest revision: 31 October 2007
Assumptions and Approximations
- The fluid is Newtonian with constant properties ( μ, ν, k , α, κ, C (^) P ).
- The flow is laminar rather than transitional or turbulent.
- The fluid is nearly incompressible – either an incompressible liquid or an ideal gas at very low Mach numbers.
Differential Equations of Motion for Nearly Incompressible Flow (general review)
- Conservation of mass : 0
i
i
u
x
- Momentum equation :
2 i i i i j i j i j j
Du u u (^) p u ρ ρ u ρg μ Dt t x x x x
- Conservation of energy (first law) : For incompressible liquid:
2
p i i
DT T
ρC k Dt x x
, where φ = 2 μ e eij ij.
For ideal gas at very low Ma :
2
p i i
DT T
ρC k Dt x x
, or
2
i i
DT T
κ Dt x x
p
k κ ρC
≡ = thermal diffusivity).
- Vorticity equation :
2 k k k j j j j
Dω u ω ω ν Dt x x x
Differential Equations of Motion for Nearly Incompressible Flow with Buoyancy
- Boussinesq approximation (See Kundu, Section 4.18): Assume that changes in density ρ are negligible
everywhere except in the gravity term (buoyancy), where we let ρ = ρ 0 ⎡ −⎣ 1 α( T − T 0 )⎤⎦ where α is the thermal
expansion coefficient,
p
ρ α ρ T
(for an ideal gas,
T
α = ), and ρ 0 is a reference density corresponding to
reference temperature T 0. ( T is assumed to vary only slightly from T 0 , so that density is nearly constant, but does
lead to buoyancy in the flow.) The density is assumed to equal ρ 0 in all other terms in the equation.
- The continuity and vorticity equations are the same as above, since density does not appear in these equations.
- The momentum equation becomes ( )
2
i i i i j i j i j j
Du u u p u ρ ρ u ρ α T T g μ Dt t x x x x
⎜ ⎟ ⎣^ ⎦
or ( )
2
0 0
i i i i j i j i j j
Du u u p u u α T T g ν Dt t x ρ x x x
⎜ ⎟ ⎣^ ⎦
, where
0
μ ν
- The energy equation then becomes: For incompressible liquid :
2
0 p i i
DT T
ρ C k Dt x x
For ideal gas at very low Mach number :
2
i i
DT T
κ Dt x x
, where
0 p
k κ ρ C
≡ (^) & viscous dissipation is negligible.
Solution Technique for Nearly Incompressible Laminar Flow without Buoyancy
- Write the continuity and momentum equations. ( Note : The energy equation is uncoupled if there is no buoyancy. If
buoyancy is important, the energy equation must be solved simultaneously with mass and momentum.)
- Simplify the equations as much as possible (cross off zero terms, etc. – always justify your simplifications).
- Solve for u (^) i and p. (This step will generate some constants from the integration).
- Apply BCs (on u (^) i and p ) to solve for the unknown constants. (Now u (^) i and p are known everywhere.)
- Write the energy equation.
- Simplify as much as possible.
- Solve for T. (This step will generate some constants from the integration).
- Apply BCs (on T ) to solve for the unknown constants. (Now T is known everywhere, and we are finished.)