CH7: Eq. of Frictionless Flow

Question 1

Nonviscous or inviscid flow requires the absence of what two phenomena?

Answer 1

The flow must be:
1. Frictionless
2. Non-conducting (no currents)

Elements of GD, Roshko, Pg. 178

Question 2

True or False
Vorticity cannot exist in one-dimensional flow.

Answer 2

True.

Elements of GD, Roshko, Pg. 178

Question 3

Stress is a tensor of ______ order.

Answer 3

2nd

Elements of GD, Roshko, Pg. 179

Question 4

A vector is a tensor of ______ order.

Answer 4

First.

Elements of GD, Roshko, Pg. 179

Question 5

Differentiation of a tensor yields a tensor of _______ order.

Answer 5

one higher

Elements of GD, Roshko, Pg. 179

Question 6

True or False
The gradient of a scaler quantity yields a vector.

Answer 6

True.

Elements of GD, Roshko, Pg. 179

Question 7

True or False
The gradient of a vector is a tensor of third order.

Answer 7

False. The gradient of a vector yields a tensor of second order. Differentiation yields a result of one order higher.

Elements of GD, Roshko, Pg. 179

Question 8

Divergence of a vector is a ________.

Answer 8

Scaler.

Elements of GD, Roshko, Pg. 180

Question 9

Give the mathematical definition of divergence.

Answer 9

Grad dot (x) where x is an arbitrary vector.

The result is a scaler.

Elements of GD, Roshko, Pg. 180

Question 10

State Gauss’ theorem from vector calculus.

Answer 10

For any volume V in a vector field b, the normal component (b dot n), integrated over the enclosing surface area A, is equal to the divergence (grad dot b), integrated over the volume.

Elements of GD, Roshko, Pg. 180

Question 11

What is the physical principle upon which the continuity equation is derived?

Answer 11

The flux of matter through a fixed control surface is constant.

Elements of GD, Roshko, Pg. 180

Question 12

In deriving the continuity equation, the surface area A must be __________.

Answer 12

Closed.

Elements of GD, Roshko, Pg. 180

Question 13

In deriving the continuity equation, the component of velocity which carries matter through the surface is defined by _________.

Hint: What is the mathematical expression that defines the component that travels normal to the surface.

Answer 13

u dot n

Where:
u = the velocity vector
n = the unit normal to the surface

Elements of GD, Roshko, Pg. 180

Question 14

In the continuity equation, the non-stationary term is due to ____________________.

Hint: Non-stationary implies time dependant or unsteady term(s).

Answer 14

The fact that the fluid density in the control volume changes if the flow is non-stationary.

Elements of GD, Roshko, Pg. 181

Question 15

What is the convective term’s physical significance in the continuity equation?

Answer 15

This term expresses the fact that the flow carries mass into and out of the control volume.

Extra Notes: The fluid which flows into and out of the control volume transports not only mass but also various characteristics associated with the fluid, such as momentum, energy, entropy, etc.

Elements of GD, Roshko, Pg. 181 - 182

Question 16

What is the primary use of gauss’s theorem in deriving the equations of motion?

Answer 16

Gauss’s theorem is used to rewrite surface integrals as volume integrals.

This is helpful because using control volumes to derive the equations is relatively straightforward.

Elements of GD, Roshko, Pg. 182

Question 17

True or False
By the momentum equation, the net force acting on the fluid in the control volume is equal to the rate of change in momentum of the fluid in the control volume.

Answer 17

True

Elements of GD, Roshko, Pg. 182

Question 18

True or False
Under steady conditions, momentum cannot accumulate in a control volume.

Answer 18

True.

Elements of GD, Roshko, Pg. 183

Question 19

What are the two kinds of forces involved in the derivation of the momentum equation?

Answer 19

Surface and volume forces.

Elements of GD, Roshko, Pg. 183

Question 20

In general, the surface forces are due to _______________.

Answer 20

Whatever medium is adjacent to the surface area, for example, a solid wall or simply the adjacent fluid.

Elements of GD, Roshko, Pg. 183

Question 21

In a non-viscous flow, what is the only possible surface force?

Answer 21

Normal Pressure forces.

Elements of GD, Roshko, Pg. 183

Question 22

True or False
In non-viscous flow, there are tangential forces adjacent to a surface that accompany normal pressure forces.

Answer 22

False. There are only normal pressure forces in non-viscous flow.

Elements of GD, Roshko, Pg. 183

Question 23

Give some examples of volume or body forces?

Answer 23

1. Inertial forces
2. Gravitational forces
3. Electromagnetic forces

Elements of GD, Roshko, Pg. 183

Question 24

Body forces are proportional to _____.

Answer 24

Mass

Elements of GD, Roshko, Pg. 183

Question 25

The energy law, applied to a given fluid in a control volume can be expressed as ________ + _________ = ___________.

Answer 25

heat added + work done on the fluid = increase in energy Elements of GD, Roshko, Pg. 185

Question 26

For a flowing fluid through a control volume the rate of change of energy can be written as ___________ + ____________ = __________.

Answer 26

Rate of heat addition + rate of work done on the fluid = rate of increase in energy of the fluid. Elements of GD, Roshko, Pg. 186

Question 27

What symbol denotes the rate of heat addition per unit mass?

Answer 27

q Elements of GD, Roshko, Pg. 186

Question 28

True or False The heat addition per unit mass (q) only includes the heat that is added externally and is not already latent in the fluid.

Answer 28

True Elements of GD, Roshko, Pg. 186

Question 29

True or False The heat released by a transformation of the fluid is not included in q, but heat absorbed from external radiation is included.

Answer 29

True Elements of GD, Roshko, Pg. 186

Question 30

The heat addition per unit mass (q) is a _______ term, and does not include heat that is transferred by _________ from one part of the fluid to the other.

Answer 30

Volume Conduction Elements of GD, Roshko, Pg. 186

Question 31

Heat that is transferred by conduction from one part of the fluid to the other is a ______ term.

Answer 31

Surface Elements of GD, Roshko, Pg. 186

Question 32

The rate of work on the fluid is due to the _______ forces and _______.

Answer 32

volume pressure (surface) Note: For inviscid flow, only pressure forces are present. Elements of GD, Roshko, Pg. 186

Question 33

The rate at which the energy is changing inside a control volume has a ________ and a _________ part.

Answer 33

non-stationary (unsteady time dependant) convective Elements of GD, Roshko, Pg. 186

Question 34

True or False Any characteristic or property associated with the fluid may be expressed as a field.

Answer 34

True Elements of GD, Roshko, Pg. 186

Question 35

According to the Eulerian derivative, the rate of change of any characteristic for a particle fluid is due to two effects. What are these two effects?

Answer 35

Convective and nonstationary effects (unsteady time dependant effects) Elements of GD, Roshko, Pg. 187

Question 36

Euler's equation is by definition, equivalent to the differential ___________ equation.

Answer 36

Momentum Elements of GD, Roshko, Pg. 188

Question 37

Whenever an extensive property such as the momentum per unit volume is associated with an intensive property such as the velocity, the continuity equation may be ___________.

Answer 37

Subtracted out. Elements of GD, Roshko, Pg. 188

Question 38

For a conservative system, the kinetic energy is interchangeable with _______________.

Answer 38

Work due to pressure (surface) and body forces. Elements of GD, Roshko, Pg. 189

Question 39

For an inviscid flow in chemical equilibrium, if there is no heat addition (q = 0), the changes in the state of the fluid particles are _____________.

Answer 39

Isentropic Elements of GD, Roshko, Pg. 189

Question 40

True or False With regard to the energy equation, the effect of friction and conduction is to create dissipation terms which are always negative.

Answer 40

False, dissipation terms are always positive because they contribute to entropy production. Elements of GD, Roshko, Pg. 189

Question 41

In steady flow particle paths coincide with _____________.

Answer 41

Streamlines Elements of GD, Roshko, Pg. 190

Question 42

In steady flow, entropy along streamlines is __________.

Answer 42

Constant Elements of GD, Roshko, Pg. 190

Question 43

For steady adiabatic flow, the total enthalpy is the same at all ______________ of a streamline.

Answer 43

Equilibrium sections Elements of GD, Roshko, Pg. 190

Question 44

Both entropy and total enthalpy are conserved along streamlines, provided what conditions are met?

Answer 44

The flow is steady, and it is also frictionless, non-conducting, and adiabatic (isentropic). Elements of GD, Roshko, Pg. 191

Question 45

What is the natural coordinate system?

Answer 45

A coordinate system in which one coordinate lies along a streamline and the other coordinate is orthogonal to the streamline. Elements of GD, Roshko, Pg. 191

Question 46

True or False In the natural coordinate system, the coordinates are said to be curvilinear.

Answer 46

True Elements of GD, Roshko, Pg. 191

Question 47

What is a stream tube?

Answer 47

A region in a flow which is bounded by two streamlines. Elements of GD, Roshko, Pg. 191 & 192

Question 48

What is shown by Crocco's Theorem?

Answer 48

How entropy varies normal to streamlines. Elements of GD, Roshko, Pg. 193

Question 49

If entropy is the same on different streamlines, the flow is said to be ____________ throughout.

Answer 49

Isentropic Elements of GD, Roshko, Pg. 193

Question 50

True or False A flow in which the entropy on different streamlines is the same must be frictionless.

Answer 50

True. Elements of GD, Roshko, Pg. 193

Question 51

Vorticity is directly related to the ____________ across streamlines.

Answer 51

entropy gradient Elements of GD, Roshko, Pg. 193

Question 52

Zero vorticity implies uniform _________ provided h0 is uniform.

Answer 52

entropy Elements of GD, Roshko, Pg. 193

Question 53

What is the equation that relates vorticity with angular velocity?

Answer 53

zeta = 2*omega Where: zeta = vorticity omega = angular velocity Elements of GD, Roshko, Pg. 195

Question 54

True or False In general, there are two components of vorticity.

Answer 54

False. There are three. Elements of GD, Roshko, Pg. 195

Question 55

A flow is irrotational only if ________ are zero.

Answer 55

All three components of vorticity. Elements of GD, Roshko, Pg. 196

Question 56

Irrotational flow is also called _______ flow.

Answer 56

Potential Elements of GD, Roshko, Pg. 197

Question 57

True or False According to Crocco's theorem, adiabatic, irrotational flows are non-isentropic.

Answer 57

False, they are Isentropic. Elements of GD, Roshko, Pg. 197

Question 58

True or False All potential flows will satisfy the Laplace Equation.

Answer 58

True. Elements of GD, Roshko, Pg. 199

Question 59

For a perfect gas, vorticity is a measure of the variation of ___________________.

Answer 59

Stagnation pressure across streamlines. Recall that stagnation pressure is directly related to the change in entropy (or the entropy gradient). Chapter 7 lectures slides, Pg 13

Question 60

Vorticity is a _______ order tensor.

Answer 60

2nd Chapter 7 lecture slides, Pg 17

Question 61

True or False Circulation is a scaler.

Answer 61

True Chapter 7 lecture slides, Pg 17

Question 62

Use of the ____________ theorem allows circulation to be related to 3D vorticity.

Answer 62

Stokes Chapter 7 lecture slides, Pg 17

Question 63

True or False Vorticity is considered a pseudo vector.

Answer 63

True. Chapter 7 lecture slides, Pg 18