Exercises 4

Fluid Mechanics
Exercise sheet 4 – Differential analysis
last edited May 28, 2015
These lecture notes are based on textbooks by White [4], Çengel & al.[6], and Munson & al.[8].
Except otherwise indicated, we assume that fluids are Newtonian, and that:
ρwater = 1 000 kg m −3 ; p atm. = 1 bar; ρatm. = 1,225 kg m −3 ; µatm. = 1,5 · 10 −5 N s m −2 ;
д = 9,81 m s −2 . Air is modeled as a perfect gas (R air = 287 J K −1 kg −1 ; γair = 1,4).
Continuity equation:
1 Dρ ~ ~
+ ∇ ·V = 0
ρ Dt
(4/17)
Navier-Stokes equation for incompressible flow:
ρ
4.1
DV~
~ + µ∇
~ 2V~
= ρ~
д − ∇p
Dt
(4/37)
Revision questions
For the continuity equation (eq. 4/17), and then for the incompressible Navier-Stokes
equation (eq. 4/37),
1. Write out the equation in its fully-developed form in three Cartesian coordinates;
2. State in which flow conditions the equation applies.
Also, in order to revise the notion of substantial derivative:
3. Describe a situation in which the substantial derivative of a property is non-zero
although the fluid property is independent of time.
4. Describe a situation in which the substantial derivative of a property is zero
although the time rate of change of this property is non-zero.
4.2
Acceleration field
Çengel & al. [6] E4-3
A flow is described with the velocity field V~ = (0,5 + 0,8x)~i + (1,5 − 0,8y)~j (in si units).
What is the acceleration measured by a probe positioned at (2; 2; 2) at t = 3 s ?
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4.3
Volumetric dilatation rate
der. Munson & al. [8] 6.4
A flow is described by the following field (in si units):
u = x 3 + y2 + z
v = xy + yz + z 3
w = −4x 2z − z 2 + 4
What is the volumetric dilatation rate field? What is the value of this rate at {2;2;2}?
4.4
Incompressibility
Çengel & al. [6] 9-28
Does the vector field V~ = (1,6 + 1,8x)~i + (1,5 − 1,8y)~j satisfy the continuity equation for
two-dimensional incompressible flow?
4.5
Missing components
Munson & al. [8] E6.2 + Çengel & al. [6] 9-4
Two flows are described by the following fields:
u1 = x 2 + y2 + z 2
v 1 = xy + yz + z
w1 = ?
u 2 = ax 2 + by 2 + cz 2
v2 = ?
w 2 = axz + byz 2
What must w 1 and v 2 be so that these flows be incompressible?
4.6
Acceleration field
White [4] E4.1
Given the velocity field V~ = (3t)~i + (xz)~j + (ty 2 )k~ (si units), what is the acceleration field,
and what is the value measured at {2;4;6} and t = 5 s?
4.7
Vortex
Çengel & al. [6] 9.27
A vortex is modeled with the following two-dimensional flow:
y
+ y2
x
v = −C 2
x + y2
u = C
x2
Verify that this field satisfies the continuity equation for incompressible flow.
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4.8
Pressure fields
Çengel & al. [6] E9-13, White [4] 4.32 & 4.34
We consider the four (separate and independent) incompressible flows below:
V~1 = (ax + b)~i + (−ay + cx)~j
V~2 = (2y)~i + (8x)~j
V~3 = (ax + bt)~i + (cx 2 + ey)~j
y
x
V~4 = U0 1 + ~i − U0 ~j
L
L
The influence of gravity is neglected on the first three fields.
Does a function exist to describe the pressure field of each of these flows, and if so, what
is it?
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Answers
4.1
1) For continuity, use eqs. 4/3 and 4/6 in eq. (4/17). For Navier-Stokes, see eqs. 4/38,
4/39 and 4/40 p. 70;
2) Read §4.4 p. 63 for continuity, and
§4.5.2 p.68 for Navier-Stokes;
3) and 4) see §4.3.2 p. 61.
DV~
Dt
4.3
= (0,4 + 0,64x )~i + (1,2 − 0,64y)~j . At the probe it takes the value 1,68~i − 0,08~j
(length 1,682 m s−2 ).
~ · V~ = −x 2 + x − z; thus at the probe it takes the value ∇
~ · V~
∇
= −4 s−1 .
4.4
Apply eq. (4/19) to V~ : the answer is yes.
4.5
1) Applying eq. (4/19): w 1 = −3xz − 21 z 2 + f (x ,y,t) ;
2) idem, v 2 = −3axy − bzy 2 + f (x ,z,t) .
4.2
4.6
probe
DV~
Dt
~ At the probe it takes the value 3~i −250~j +490k.
~
= (3)~i +(3z +y 2x)t ~j +(y 2 +2xyzt)k.
4.7
Apply eq. (4/19) to V~ to verify incompressibility.
4.8
Keep in mind that the unknown functions f may include a constant (initial)
value p 0 .
h
i
1) p = −ρ abx + 21 a 2x 2 + bcy + 12 a 2y 2 + f (t) ;
∂p ∂ ∂p
2) p = −ρ 8x 2 + 8y 2 + f (t);
3) ∂x∂ ∂y , ∂y
∂x , thus we cannot
describe thepressure
with a mathematical function;
4) p = −ρ
U02
L
x+
x2
2L
+
y2
2L
− дx x − дyy + f (t) .
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