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Professor Fred Stern Fall 2013 (^1)
Chapter 4: Fluids Kinematics
4.1 Velocity and Description Methods
Primary dependent variable is fluid velocity vector V = V ( r ); where r is the position vector
If V is known then pressure and forces can be determined using techniques to be discussed in subsequent chapters.
Consideration of the velocity field alone is referred to as flow field kinematics in distinction from flow field dynamics (force considerations).
Fluid mechanics and especially flow kinematics is a geometric subject and if one has a good understanding of the flow geometry then one knows a great deal about the solution to a fluid mechanics problem.
Consider a simple flow situation, such as an airfoil in a wind tunnel:
r xiˆyˆjzkˆ
V r t ( , ) ui ˆ^ vj ˆ wk ˆ
x
r
U = constant
Professor Fred Stern Fall 2013 (^2)
Velocity: Lagrangian and Eulerian Viewpoints
There are two approaches to analyzing the velocity field: Lagrangian and Eulerian
Lagrangian: keep track of individual fluids particles (i.e., solve F = Ma for each particle) Say particle p is at position r 1 (t 1 ) and at position r 2 (t 2 ) then,
̂ ̂ ̂ ̂ ̂ ̂
Of course the motion of one particle is insufficient to describe the flow field, so the motion of all particles must be considered simultaneously which would be a very difficult task. Also, spatial gradients are not given directly. Thus, the Lagrangian approach is only used in special circumstances.
Eulerian: focus attention on a fixed point in space
̂̂̂
In general, ( )̂̂̂ (^) ⏟
where, ( ), ( ), ( )
Professor Fred Stern Fall 2013 (^4)
4.2 Acceleration Field and Material Derivative
The acceleration of a fluid particle is the rate of change of its velocity.
In the Lagrangian approach the velocity of a fluid particle is a function of time only since we have described its motion in terms of its position vector.
( ) ̂ ( ) ̂ ( ) ̂
̂ ̂ ̂
̂ ̂ ̂
In the Eulerian approach the velocity is a function of both space and time such that,
( ) ̂ ( ) ̂ ( ) ̂
where (^ )^ are velocity components in (^ ) directions, and (^ )^ ( )^ since we must follow the particle in evaluating ⁄.
Professor Fred Stern Fall 2013 (^5)
where ( ) are not arbitrary but assumed to follow a
fluid particle, i.e.
Similarly for & ,
Professor Fred Stern Fall 2013 (^7) Example: Flow through a converging nozzle can be approximated by a one dimensional velocity distribution u = u(x). For the nozzle shown, assume that the velocity varies linearly from u = Vo at the entrance to u = 3Vo at the exit. Compute the acceleration
Dt
DV
as a function of x.
Evaluate Dt
DV
at the entrance
and exit if Vo = 10 ft/s and L =1 ft.
We have V u(x)iˆ, ax x
u u Dt
Du
(^)
^ 1
L
2 x x V V L
2 V
u (x) o o o
L
2 V
x
u (^0)
^ 1
L
2 x L
2 V
a
2 o x
@ x = 0 ax = 200 ft/s^2
@ x = L ax = 600 ft/s^2
u = Vo
y
Assume linear variation between inlet and exit
u(x) = mx + b u(0) = b = Vo m = L
2 V L
3 V V x
u (^) o o o
Professor Fred Stern Fall 2013 (^8)
Additional considerations: Separation, Vortices,
Turbulence, and Flow Classification
We will take this opportunity and expand on the material provided in the text to give a general discussion of fluid flow classifications and terminology.
- One-, Two-, and Three-dimensional Flow
1D: V =u(y)iˆ
2D: V =u( x,y)iˆv(x,y)ˆj
3D: V = V(x) = u (x,y,z)iˆv(x,y,z)ˆjw(x,y,z)kˆ
- Steady vs. Unsteady Flow V = V(x,t) unsteady flow
V = V(x) steady flow
- Incompressible and Compressible Flow
0 Dt
D
incompressible flow
representative velocity
Ma = c
V
speed of sound in fluid
Professor Fred Stern Fall 2013 (^10)
are usually irrotational. Inviscid, irrotational, incompressible flow is referred to as ideal-flow theory.
- Laminar vs. Turbulent Viscous Flows Laminar flow = smooth orderly motion composed of thin sheets (i.e., laminas) gliding smoothly over each other
Turbulent flow = disorderly high frequency fluctuations superimposed on main motion. Fluctuations are visible as eddies which continuously mix, i.e., combine and disintegrate (average size is referred to as the scale of turbulence). Reynolds decomposition u uu(t)
mean turbulent fluctuation motion
usually u(.01-.1) u, but influence is as if increased by 100-10,000 or more.
Professor Fred Stern Fall 2013 (^11)
Example: Pipe Flow (Chapter 8 = Flow in Conduits) Laminar flow:
dx
dp 4
R r u(r)
2 2
u(y),velocity profile in a paraboloid
Turbulent flow: fuller profile due to turbulent mixing extremely complex fluid motion that defies closed form analysis.
Turbulent flow is the most important area of motion fluid dynamics research.
The most important nondimensional number for describing fluid motion is the Reynolds number (Chapter 8)
Re =
VD VD
V = characteristic velocity D = characteristic length
Professor Fred Stern Fall 2013 (^13)
Flow Field Regions (high Re flows)
Important features:
- low Re viscous effects important throughout entire fluid domain: creeping motion
- high Re flow about streamlined body viscous effects confined to narrow region: boundary layer and wake
- high Re flow about bluff bodies: in regions of adverse pressure gradient flow is susceptible to separation and viscous-inviscid interaction is important
- Separated vs. Unseparated Flow
Flow remains attached Streamlined body w/o separation
Bluff body Flow separates and creates the region of reverse flow, i.e. separation
viscous force
Vc inertia force Re
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4.3 Basic Control-Volume Approach and RTT
Professor Fred Stern Fall 2013 (^16)
Reynolds Transport Theorem (RTT)
Need relationship between B sys
dt
d
and changes in
Bcv CV dm CV d .
1 = time rate of change of B in CV =
CV
d dt
d dt
dBcv
2 = net outflux of B from CV across CS =
SYS R CV CS
dB d (^) d V n dA dt dt
General form RTT for moving deforming control volume
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Special Cases:
1) Non-deforming CV
SYS R CV CS
dB d V n dA dt t
(^)
2) Fixed CV
SYS CV CS
dB d V n dA dt t
(^)
Gauss’s Theorem:
CV CS
^ b d^^ ^ b n dA
SYS
CV
dB V d dt t
^
Since CV fixed and arbitrary lim d 0 gives differential eq.
3) Steady Flow: 0
t
4) Uniform flow across discrete CS (steady or
unsteady)
CS CS
^ ^ V^ ^ n dA^ ^ V^ n dA (- inlet, + outlet)
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3. Incompressible fluid ( = constant)
CS CV
V dA d dV dt ^ ^ conservation of volume
4. Steady One-Dimensional Flow in a Conduit:
CS
V A 0
1 V 1 A 1 + 2 V 2 A 2 = 0
for = constant Q 1 = Q 2
