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Test 1 Number of dimensions based on x, y, and z Steady depends on t ̂
̂
̂
If Steady:
Pathline = Streakline = Streamline
Streamline:
A line that is everywhere tangent to local velocity vector
Pathline:
Actual path traversed by a fluid particle
⃑⃑ ̇ ̇
Streakline:
Particles that have pass through a specific
⃑
Test 2
point
Conservation of Mass:
For block example and really really small h:
∫ ⃑
Momentum Equation:
⃑
⃑
∫ ⃑
∫
⃑
∫ ⃑
∑⃑
Conservation of Energy:
̇
Laminar flow: Low Reynolds number, smooth flow
̇
⃑⃑
∫
⃑)
(⃑
∫
or
Unstable flow: Re = Recrit Turbulent flow: Re > Recrit ̇ ̇
√
∑
⃑⃑⃑⃑ ̇
∑
⃑
∫
⃑
∫⃑
⃑
∫
∫
For Mach < 0.3 assume incompressible
Assumptions:
Steady State: Incompressible: Uniform Velocity Ideal gas:
Subsonic jet:
⃑
̇
⃑⃑⃑⃑
Bernoulli Equation:
Test 3 Continuity Equation (Possible Incompressible Flow):
Assumptions:
Steady State: Incompressible: Inviscid
The number of dimensions is dependent on x, y, and z
Along the same Streamline Particle Acceleration: Reynolds Number (other side of page) ⃑ ⃑ Mach Number (other side of page) Pressure Gradient: Euler Number (Pressure Coefficient):
⃑ ⃑
Weber Number (for bubbles):
Irrotational Fluid: ⃑⃑
⃑⃑
Froude Number:
Rotation: ̂ ⃑⃑
̂ ̂
|
√
|
Drag Coefficient: Circulation: ∮ ⃑⃑
(
⃑⃑
For a square element: ∫(
)
)
Flow Similarity and Model Studies: 1) Geometric Similarity (Same shape with some scale factor)
Navier-Stokes Equation: Force Term has ρ*equation The other is the Acceleration Term
2) Kinematic Similarity (Velocities in same direction and scale between sizes) 3) Dynamic Similarity (Same Force Distribution):
Final (Chapter 8) Stationary Plates (
)[
]
Volumetric Flow Rate per Length: (
)
Pipes Laminar Flow (
)
(
)
(
)
̇ ̇ If Laminar:
If Turbulent: ∑ ( )
( )
For
For
*
and
(
⁄
)+
Cavitation is Low-Pressure Boiling and most likely to occur at max velocity
