Solutions Manual Chapter 10 Approximate Solutions Of

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Solutions Manual, Chapter 10 – Approximate Solutions of the N-S Equation

Chapter 10

Approximate

Solutions of the

Navier-Stokes

Equation

General and Introductory Problems, Modified Pressure, Fluid Statics

10-1C

Solution We are to discuss the difference between an “exact” solution and an

approximate solution of the Navier-Stokes equation.

Analysis In an “exact” solution, we begin with the full Navier-Stokes

equation. As we solve the problem, some terms may drop out due to the specified

geometry or other simplifying assumptions in the problem. In an approximate

solution, we eliminate some terms in the Navier-Stokes equation right from the

start. In other words, we begin with a reduced or simplified approximate form of the

equation.

Discussion The approximations are based on the class of flow problem and/or the

region in which such approximations are appropriate (e.g. irrotational, boundary

layer, etc.).

10-1

Solutions Manual, Chapter 10 – Approximate Solutions of the N-S Equation

10-2C

Solution We are to label regions in a flow field where certain approximations

are likely to be appropriate.

Assumptions 1 The flow is incompressible. 2 The flow is steady in the mean (we

ignore the unsteady flow field close to the rotating blades).

Irrotational

Static Static

Boundary layer

Full Navier-

Stokes

FIGURE 1

Regions of appropriate approximations for

the flow produced by a box fan sitting on the

floor of a large room.

Analysis A boundary layer grows along the floor, both upstream and

downstream of the fan. The flow upstream of the fan is largely irrotational except

very close to the floor. The air is nearly static far upstream and far above the fan.

Downstream of the fan, the flow is most likely swirling and turbulent, and none of the

approximations are expected to be appropriate there. In other words, the full Navier-

Stokes equation must be solved in that region. We sketch all these regions in Fig. 1.

Discussion The regions sketched in Fig. 1 are not well defined, nor are they

necessarily to scale.

10-3C

Solution We are to discuss the role of nondimensionalization of the Navier-

Stokes equations.

Analysis When we properly nondimensionalize the Navier-Stokes equation, all

the terms are re-written in the form of some nondimensional parameter times a

quantity of order unity. Thus, we can simply compare the orders of magnitude of

the nondimensional parameters to see which terms (if any) can be ignored because

they are very small compared to other terms. For example, if the Strouhal number is

much smaller than the Euler number, we can ignore the term that contains the

Strouhal number, but must retain the term that contains the Euler number.

Discussion This method works only if the characteristic scales of the problem

(length, speed, frequency, etc.) are chosen properly.

10-4C

Solution We are to discuss the most significant danger that arises with an

approximate solution, and we are to come up with an example.

Analysis The danger of an approximate solution of the Navier-Stokes equation

is this: If the approximation is not appropriate to begin with, our solution will be

incorrect – even if we perform all the mathematics correctly. There are many

examples. For instance, we may assume that a boundary layer exists in a region of

flow. However, if the Reynolds number is not large enough, the boundary layer is too

thick and the boundary layer approximations break down. Another example is that we

may assume a fluid statics region, when in reality there are swirling eddies in that

region. The unsteady motion of the eddies makes the problem unsteady and dynamic

– the approximation of fluid statics would be inappropriate.

Discussion When you make an approximation and solve the problem, it is best to

go back and verify that the approximation is appropriate.

10-2

Solutions Manual, Chapter 10 – Approximate Solutions of the N-S Equation

10-5C

Solution We are to discuss the criteria used to determine whether an

approximation of the Navier-Stokes equation is appropriate or not

Analysis We determine if an approximation is appropriate by comparing the

orders of magnitude of the various terms in the equations of motion. If the

neglected terms are negligibly small compared to other terms, then the approximation

is appropriate. If not, then it is not appropriate to neglect those terms.

Discussion It is important that the proper scales be used for the

nondimensionalization of the equation. Otherwise, the order of magnitude analysis

may be incorrect.

10-6C

Solution We are to discuss the physical significance of the four

nondimensional parameters in the nondimensionalized incompressible Navier-Stokes

equation.

Analysis The four parameters are discussed individually below:

 Strouhal number: St is the ratio of some characteristic flow time to some

period of oscillation. If St << 1, the oscillation period is very large compared

to the characteristic flow time, and the problem is quasi-steady; the unsteady

term in the Navier-Stokes equation may be ignored. If St >> 1, the

oscillation period is very short compared to the characteristic flow time, and

the unsteadiness dominates the problem; the unsteady term must remain.

 Euler number: Eu is the ratio of a characteristic pressure difference to a

characteristic pressure due to fluid inertia. If Eu << 1, pressure gradients are

very small compared to inertial pressure, and the pressure term can be

neglected in the Navier-Stokes equation. If Eu >> 1, the pressure term is

very large compared to the inertial term, and must remain in the equation.

 Froude number: Fr is the ratio of inertial forces to gravitational forces.

Note that Fr appears in the denominator of the nondimensionalized Navier-

Stokes equation. If Fr << 1, gravitational forces are very large compared to

inertial forces, and the gravity term must remain in the Navier-Stokes

equation. If Fr >> 1, gravitational forces are negligible compared to inertial

forces, and the gravity term in the Navier-Stokes equation can be ignored.

 Reynolds number: Re is the ratio of inertial forces to viscous forces. Note

that Re appears in the denominator of the nondimensionalized Navier-Stokes

equation. If Re << 1, viscous forces are very large compared to inertial

forces, and the viscous term must remain. (In fact, it may dominate the other

terms, as in creeping flow). If Re >> 1, viscous forces are negligible

compared to inertial forces, and the viscous term in the Navier-Stokes

equation can be ignored. Note that this applies only to regions outside of

boundary layers, because the characteristic length scale for a boundary layer

is generally much smaller than that for the overall flow.

Discussion You must keep in mind that the approximations discussed here are

appropriate only in certain regions of the flow field. In other regions of the same flow

field, different approximations may apply.

10-3

Solutions Manual, Chapter 10 – Approximate Solutions of the N-S Equation

10-7C

Solution We are to discuss the criterion for using modified pressure.

Analysis Modified pressure can be used only when there are no free surface

effects in the problem.

Discussion Modified pressure is simply a combination of thermodynamic

pressure and hydrostatic pressure. It turns out that if there are no free surface effects,

the hydrostatic pressure component is independent of the flow pressure component,

and these two can be separated.

10-8C

Solution We are to discuss which nondimensional parameter is eliminated by

use of the modified pressure.

Analysis Modified pressure effectively combines the effects of actual pressure

and gravity. In the nondimensionalized Navier-Stokes equation in terms of modified

pressure, the Froude number disappears. The reason Froude number is eliminated

is because the gravity term is eliminated from the equation.

Discussion Keep in mind that we can employ modified pressure only for flows

without free surface effects.

10-9

Solution We are to plug the given scales for this flow problem into the

nondimensionalized Navier-Stokes equation to show that only two terms remain in

the region consisting of most of the tank.

Assumptions 1 The flow is incompressible. 2 d << D. 3 D is of the same order of

magnitude as H.

Analysis The characteristic frequency is taken as the inverse of the

characteristic time, f = 1/tdrain. The Strouhal number is thus

Strouhal number:

drain

St ~ 1

fL H

VtV

== (1)

St is of order of magnitude 1 since the order of magnitude of tdrain is H/V. The Euler

number is

Euler number:

24

jet

0

222

Eu ~ ~

V

PP

4

g

HD

VVV

ρ

ρρ

∞

−

== d

(2)

where we have used the order of magnitude estimate that jet gHV~ . We have also

used conservation of mass, namely Vjetd2 = VtankD2. Similarly, the Froude number is

10-4