Category Aerodynamics for Engineering Students

Few physically realizable exact solutions of the Navier-Stokes equations exist. Even fewer are of much interest in Engineering. Here we will present the two simplest solutions, namely Couette flow (simple shear flow) and plane Poiseuille flow (channel flow). These are useful for engineering applications, although not for the aerody­namics of wings and bodies. The third exact solution represents the flow in the vicinity of a stagnation point. This is important for calculating the flow around wings and bodies. It also illustrates a common and, at first sight, puzzling feature. Namely, that if the dimensionless Navier-Stokes equations can be reduced to an ordinary differential equation, this is regarded as tantamount to an exact solution. This is because the essentials of the flow field can be represented in terms of one or two curves plotted on a single graph. Also numerical solutions to ordinary differ­ential equations can be obtained to any desired accuracy.

1.9.1 Couette flow – simple shear flow

This is the simplest exact solution. It corresponds to the flow field created between two infinite, plane, parallel surfaces; the upper one moving tangentially at speed UT, the lower one being stationary (see Fig. 2.30). Since the flow is steady and two­dimensional, derivatives with respect to c and l are zero, and ir = 0. The streamlines

Fig. 2.30 are parallel to the x axis, so v = 0. Therefore Eqn (2.93) implies du/dx = 0, i. e. и is a function only of y. There is no external pressure field, so Eqn (2.92a) reduces to

r2

= 0 implying и = Cy + C2 (2.106)

where Ci and C2 are constants of integration, и = 0 and Uj when у = 0 and h respectively, so Eqn (2.106) becomes

и=иТт = – У (2.107)

h Ц

where r is the constant viscous shear stress.

This solution approximates well the flow between two concentric cylinders with the inner one rotating at fixed speed, provided the clearance is small compared with the cylinder’s radius, R. This is the basis of a viscometer – an instrument for measuring viscosity, since the torque required to rotate the cylinder at constant speed uj is proportional to r which is given by nuR/h. Thus if the torque and rotational speed are measured the viscosity can be determined.

Restricting our derivation to two-dimensional flow, Eqn (2.87) with (2.72a) and (2.73) gives

So the right-hand side of the momentum Eqns (2.66a) becomes

The right-hand side of (2.66b) can be dealt with in a similar way. Thus the momen­tum equations (2.66a, b) can be written in the form

(2.92a)

(2.92b)

This form of the momentum equations is known as the Navier-Stokes equations for two-dimensional flow. With the inclusion of the continuity equation

du dv

d7x + d-y = Q we now have three governing equations for three unknown flow variables u, v, p.

The Navier-Stokes equations for three-dimensional incompressible flows are given below:

1.9 Properties of the Navier-Stokes equations

At first sight the Navier-Stokes equations, especially the three-dimensional version, Eqns (2.95), may appear rather formidable. It is important to recall that they are nothing more than the application of Newton’s second law of motion to fluid flow.

For example, the left-hand side of Eqn (2.95a) represents the total rate of change of the x component of momentum per unit volume. Indeed it is often written as:

Du D д д д d

’Di where msm+uTx+%+wdz

is called the total or material derivative. It represents the total rate of change with time following the fluid motion. The left-hand sides of Eqns (2.95b, c) can be written in a similar form. The three terms on the right-hand side represent the x components of body force, pressure force and viscous force respectively acting on a unit volume of fluid.

The compressible versions of the Navier-Stokes equations plus the continuity equation encompass almost the whole of aerodynamics. To be sure, applications involving combustion or rarified flow would require additional chemical and phys­ical principles, but most of aerodynamics is contained within the Navier-Stokes equations. Why, then, do we need the rest of the book, not to mention the remaining vast, ever-growing, literature devoted to aerodynamics? Given the power of modem computers, could we not merely solve the Navier-Stokes equations numerically for any aerodynamics application of interest? The short answer is no! Moreover, there is no prospect of it ever being possible. To explain fully why this is so is rather difficult. We will, nevertheless, attempt to give a brief indication of the nature of the problem.

Let us begin by noting that the Navier-Stokes equations are a set of partial differential equations. Few analytical solutions exist that are useful in aerodynamics. (The most useful examples will be described in Section 2.10.) Accordingly, it is essential to seek approximate solutions. Nowadays, it is often possible to obtain very accurate numerical solutions by using computers. In many respects these can be regarded almost as exact solutions, although one must never forget that computer­generated solutions are subject to error. It is by no means simple to obtain such numerical solutions of the Navier-Stokes equations. There are two main sources of difficulty. First, the equations are nonlinear. The nonlinearity arises from the left – hand sides, i. e. the terms representing the rate of change of momentum – the so-called inertial terms. To appreciate why these terms are nonlinear, simply note that when you take a term on the right-hand side of the equations, e. g. the pressure terms, when the flow variable (e. g. pressure) is doubled the term is also doubled in magnitude. This is also true for the viscous terms. Thus these terms are proportional to the unknown flow variables, i. e. they are linear. Now consider a typical inertial term, say иди/дх. This term is plainly proportional to u2 and not u, and is therefore nonlinear. The second source of difficulty is more subtle. It involves the complex effects of viscosity.

In order to understand this second point better, it is necessary to make the Navier – Stokes equations non-dimensional. The motivation for working with non-dimen­sional variables and equations is that it helps to make the theory scale-invariant and accordingly more universal (see Section 1.4). In order to fix ideas, let us consider the air flowing at speed U00 towards a body, a circular cylinder or wing say, of length L. See Fig. 2.29. The space variables x, y, and z can be made nondimensional by dividing by

L. L/Uqc can be used as the reference time to make time non-dimensional. Thus we introduce the non-dimensional coordinates

X = x/L, Y = y/L, Z = z/L, and T = lU/L (2.97)

Uoo can be used as the reference flow speed to make the velocity components dimensionless and pU^ (c. f. Bernoulli equation Eqn (2.16)) used as the reference

U.

Fig. 2.29

pressure. (For incompressible flow, at least, only pressure difference is of significance and not the absolute value of the pressure.) This allows us to introduce the following non-dimensional flow variables:

U = u/Uoo, V = v/Ux, W = w/Uoo, and P=p/(pU20O) (2.98)

If, by writing x — XL etc. the non-dimensional variables given in Eqns (2.97) and

(2.98) are substituted into Eqns (2.94) and (2.95) with the body-force terms omitted, we obtain the Navier-Stokes equations in the form:

where the short-hand notation (2.96) for the material derivative has been used. A feature of Eqns (2.100) is the appearance of the dimensionless quantity known as the Reynolds number.

Re = ^°-h (2.Ю1)

From the manner in which it has emerged from making the Navier-Stokes equations dimensionless, it is evident that the Reynolds number (see also Section 1.4) represents the ratio of the inertial to the viscous terms (i. e. the ratio of rate of change of momentum to the viscous force). It would be difficult to overstate the significance of Reynolds number for aerodynamics.

It should now be clear from Eqns (2.99) and (2.100) that if one were to calculate the non-dimensional flow field for a given shape – a circular cylinder, for example – the overall flow pattern obtained would depend on the Reynolds number and, in the case of unsteady flows, on the dimensionless time T. The flow around a circular cylinder is a good example for illustrating just how much the flow pattern can change over a wide range of Reynolds number. See Section 7.5 and Fig. 7.14 in particular. Incidentally, the simple dimensional analysis carried out above shows that it is not always necessary to solve equations in order to extract useful information from them.

For high-speed flows where compressibility becomes important the absolute value of pressure becomes significant. As explained in Section 2.3.4 (see also Section 1.4), this leads to the appearance of the Mach number, M (the ratio of the flow speed to the speed of sound), in the stagnation pressure coefficient. Thus, when compressibility
becomes important (see Section 2.3.4), Mach number becomes a second dimension­less quantity characterizing the flow field.

The Navier-Stokes equations are deceptively simple in form, but at high Reynolds numbers the resulting flow fields can be exceedingly complex even for simple geo­metries. This is basically a consequence of the behaviour of the regions of vortical flow at high Reynolds number. Vorticity can only be created in a viscous flow and can be regarded as a marker for regions where the effects of viscosity are important in some sense.

For engineering applications of aerodynamics the Reynolds numbers are very large, values well in excess of 106 are commonplace. Accordingly, one would expect that to a good approximation one could drop the viscous terms on the right-hand side of the dimensionless Navier-Stokes Eqns (2.100). In general, however, this view would be mistaken and one never achieves a flow field similar to the inviscid one no matter how high the Reynolds number. The reason is that the regions of non-zero vorticity where viscous effects cannot be neglected become confined to exceedingly thin boundary layers adjacent to the body surface. As Re —> oc the boundary-layer thickness, 6 —► 0. If the boundary layers remained attached to the surface they would have little effect beyond giving rise to skin-friction drag. But in all real flows the boundary layers separate from the surface of the body, either because of the effects of an adverse pressure gradient or because they reach the rear of the body or its trailing edge. When these thin regions of vortical flow separate they form complex unsteady vortex-like structures in the wake. These take their most extreme form in turbulent flow which is characterized by vortical structures with a wide range of length and time scales.

As we have seen from the discussion given above, it is not necessary to solve the Navier-Stokes equations in order to obtain useful information from them. This is also illustrated by following example:

Example 2.3 Aerodynamic modelling

Let us suppose that we are interested carrying out tests on a model in a wind-tunnel in order to study and determine the aerodynamic forces exerted on a motor vehicle travelling at normal motorway speeds. In this case the speeds are sufficiently low to ensure that the effects of compressibility are negligible. Thus for a fixed geometry the flow field will be characterized only by Reynolds number.[6] In this case we can use Ux, the speed at which the vehicle travels (the air speed in the wind-tunnel working section for the model) as the reference flow speed, and L can be the width or length of the vehicle. So the Reynolds number Re = pUxLjfi. For a fixed geometry it is clear from Eqns (2.99) and (2.100) that the non-dimensional flow variables, U, V, W, and P are functions only of the dimensionless coordinates X, Y, Z, T, and the dimensionless quantity, Re. In a steady flow the aerodynamic force, being an overall characteristic of the flow field, will not depend on X, Y, Z, or T. It will, in fact, depend only on Re. Thus if we make an aerodynamic force, drag (D) say, dimensionless, by introducing a force (i. e. drag) coefficient defined as

(see Section 1.5.2 and noting that here we have used I? in place of area S) it should be clear that

Cd = F(Re) i. e. a function of Re only

If we wish the model tests to produce useful information about general characteristics of the prototype’s flow field, in particular estimates for its aerodynamic drag, it is necessary for the model and prototype to be dynamically similar, i. e. for the forces to be scale invariant. It can be seen from Eqn (2.103) that this can only be achieved provided

where suffices m and p denote model and prototype respectively.

It is not usually practicable to use any other fluid but air for the model tests. For standard wind-tunnels the air properties in the wind-tunnel are not greatly different from those experi­enced by the prototype. Accordingly, Eqn (2.104) implies that

(2.105)

Thus, if we use a 1/5-scale model, Eqn (2.105) implies that Um = 5Up. So a prototype speed of lOOkm/hr (c. 30m/s) implies a model speed of 500km/hr (c. 150m/s). At such a model speed compressibility effects are no longer negligible. This illustrative example suggests that, in practice, it is rarely possible to achieve dynamic similarity in aerodynamic model tests using standard wind-tunnels. In fact, dynamic similarity can usually only be achieved in aerody­namics by using very large and expensive facilities where the dynamic similarity is achieved by compressing the air (thereby increasing its density) and using large models.

In this example we have briefly revisited the material covered in Section 1.4. The objective was to show how the dimensional analysis of the Navier-Stokes equations (effectively the exact governing equations of the flow field) could establish more rigorously the concepts introduced in Section 1.4.

2.8.1 Relationship between rates of strain and viscous stresses

In solid mechanics the fundamental theoretical model linking the stress and strain fields is Hooke’s law that states that

and Eqn (2.87) will, accordingly, be valid.

The total amount of vortidty passing through any plane region within a flow field is called the circulation, Г. This is illustrated in Fig. 2.27 which shows a bundle of vortex tubes passing through a plane region of area A located in the flow field. The perimeter of the region is denoted by C. At a typical point P on the perimeter, the velocity vector is designated q or, equivalently, q. At P, the infinitesimal portion of C has length 6s and points in the tangential direction defined by the unit vector t (or F). It is important to understand that the region of area A and its perimeter C have no physical existence. Like the control volumes used for the application of conservation of mass and momentum, they are purely theoretical constructs.

Mathematically, the total strength of the vortex tubes can be expressed as an integral over the area A; thus

Г = її n ■ SldA

A

where n is the unit normal to the area A. In two-dimensional flow the vortidty is in the z direction perpendicular to the two-dimensional flow field in the (x, y) plane. Thus n = к (i. e. the unit vector in the z direction) and ft = £k, so that Eqn (2.80) simplifies to

* Vortidty can also be created by other agencies, such as the presence of spatially varying body forces in the flow field. This could correspond to the presence of particles in the flow field, for example.

Fig. 2.27

Circulation can be regarded as a measure of the combined strength of the total number of vortex lines passing through A. It is a measure of the vorticity flux carried through A by these vortex lines. The relationship between circulation and vorticity is broadly similar to that between momentum and velocity or that between internal energy and temperature. Thus circulation is the property of the region A bounded by control surface C, whereas vorticity is a flow variable, like velocity, defined at a point. Strictly it makes no more sense to speak of conservation, generation, or transport of vorticity than its does to speak of conservation, generation, or transport of velocity. Logically these terms should be applied to circulation just as they are to momentum rather than velocity. But human affairs frequently defy logic and aero­dynamics is no exception. We have become used to speaking in terms of conservation etc. of vorticity. It would be considered pedantic to insist on circulation in this context, even though this would be strictly correct. Our only motivation for making such fine distinctions here is to elucidate the meaning and significance of circulation. Henceforth we will adhere to the common usage of the terms vorticity and circulation.

In two-dimensional flow, in the absence of the effects of viscosity, circulation is conserved. This can be expressed mathematically as follows:

£)( d( c>C_n

or ox Oy

In view of what was written in Section 2.7.6 about the link between vorticity and viscous effects, it may seem somewhat illogical to neglect such effects in Eqn (2.82). Nevertheless, it is often a useful approximation to use Eqn (2.82).

Circulation can also be evaluated by means of an integration around the perimeter C. This can be shown elegantly by applying Stokes theorem to Eqn (2.81); thus

Г = Jjn-QdA = Jjn V x 4dA = ! q-tds (2.83)

This commonly serves as the definition of circulation in most aerodynamics text.

The concept of circulation is central to the theory of lift. This will become clear in Chapters 5 and 6.

Fig. 2.28

Example 2.2 For the rectangular region of a two-dimensional flow field depicted in Fig. 2.28, starting with the definition Eqn (2.81) of circulation, show that it can also be evaluated by means of the integral around the closed circuit appearing as the last term in Eqn (2.83).

From Eqns (2.76) and (2.81) it follows that

Referring to Section 2.4.3 where polar coordinates were introduced, the correspond­ing definition of vorticity in polar coordinates is

g, dq, 1 dqn

^ r dr r ffl

Note that consistent with its physical interpretation as rate of rotation, the units of vorticity are radians per second.

Fig. 2.26

2.7.3 Rotational and irrotational flow

It will be made clear in Section 2.8 that the generation of shear strain in a fluid element, as it travels through the flow field, is closely linked with the effects of viscosity. It is also plain from its definition (Eqn (2.76)) that vortidty is related to rate of shear strain. Thus, in aerodynamics, the existence of vortidty is associated with the effects of viscosity.* Accordingly, when the effects of viscosity can be neglected, the vortidty is usually equivalently zero. This means that the individual fluid elements do not rotate, or distort, as they move through the flow field. For incompressible flow, then, this corresponds to the state of pure translation that is illustrated in Fig. 2.26. Such a flow is termed irrotational flow. Mathematically, it is characterized by the existence of a velodty potential and is, therefore, also called potential flow. It is the subject of Chapter 3. The converse of irrotational flow is rotational flow.

Following an analogous process we can also calculate the direct strains and their corresponding rates of strain, for example

The other direct strains are obtained in a similar way; thus the rates of direct strain are given by

Thus we can introduce a rate of strain tensor analogous to the stress tensor (see Section 2.6) and for which components in two-dimensional flow can be represented in matrix form as follows:

where ( ‘ ) is used to denote a time derivative.

2.7.2 Vorticity

The instantaneous rate of rotation of a fluid element is given by (a — /3)/2 – see above. This corresponds to a fundamental property of fluid flow called the vorticity that, using Eqn (2.71), in two-dimensional flow is defined as

da dp _ dv du dr dr dx dy

In three-dimensional flow vorticity is a vector given by

dw dv du dw dv du dy dz’ dz dx ’ dx dy)

It can be seen that the three components of vorticity are twice the instantaneous rates of rotation of the fluid element about the three coordinate axes. Mathematically it is given by the following vector operation

Q = Vxv (2.78)

Vortex lines can be defined analogously to streamlines as lines that are tangential to the vorticity vector at all points in the flow field. Similarly the concept of the vortex tube is analogous to that of stream tube. Physically we can think of flow structures like vortices as comprising bundles of vortex tubes. In many respects vorticity and vortex lines are even more fundamental to understanding the flow physics than are velocity and streamlines.

Consider Fig. 2.25. This shows an elemental control volume ABCD that initially at time t = tj is square. After an interval of time 6t has elapsed ABCD has moved and distorted into A’B’C’D’. The velocities at t = /,■ at A, В and C are given by

Fig. 2.25

As they stand, the momentum Eqns (2.66) (or 2.67), together with the continuity Eqn (2.46) (or 2.47) cannot be solved, even in principle, for the flow velocity and pressure. Before this is possible it is necessary to link the viscous stresses to the velocity field through a constitutive equation. Air, and all other homogeneous gases and liquids, are closely approximated by the Newtonian fluid model. This means that the viscous stress is proportional to the rate of strain. Below we consider the distortion experi­enced by an infinitesimal fluid element as it travels through the flow field. In this way we can derive the rate of strain in terms of velocity gradients. The important flow properties, vorticity and circulation will also emerge as part of this process.

2.7.1 Distortion of fluid element in flow field

Figure 2.24 shows how a fluid element is transformed as it moves through a flow field. In general the transformation comprises the following operations:

(i) Translation – movement from one position to another.

(ii) DilationjCompression – the shape remains invariant, but volume reduces or increases. For incompressible flow the volume remains invariant from one position to another.

(iii) Distortion – change of shape keeping the volume invariant.

Distortion can be decomposed into anticlockwise rotation through angle

(a — 0)/2 and a shear of angle (a + /?)/2.

The angles a and (3 are the shear strains.

Fig. 2.24 Transformation of a fluid element as it moves through the flow field

For some applications in aerodynamics it can be an acceptable approximation to neglect the viscous stresses. In this case Eqns (2.66) simplify to

(2.68a)

(2.68b)

These equations are known as the Euler equations. In principle, Eqns (2.68a, b), together with the continuity Eqn (2.46), can be solved to give the velocity components и and v and pressure p. However, in general, this is difficult because Eqns (2.68a, b) can be regarded as the governing equations for и and v, but p does not appear explicitly in the continuity equation. Except for special cases, solution of the Euler equations can only be achieved numerically using a computer. A very special and comparatively simple case is irrotational flow (see Section 2.7.6). For this case the Euler equations reduce to a single simpler equation – the Laplace equation. This equation is much more amenable to analytical solution and this is the subject of Chapter 3.

The momentum equation for two – or three-dimensional flow embodies the applica­tion of Newton’s second law of motion (mass times acceleration = force, or rate of change of momentum = force) to an infinitesimal control volume in a fluid flow (see Fig. 2.8). It takes the form of a set of partial differential equations. Physically it states that the rate of increase in momentum within the control volume plus the net rate at which momentum flows out of the control volume equals the force acting on the fluid within the control volume.

There are two distinct classes of force that act on the fluid particles within the control volume.

(i) Body forces. Act on all the fluid within the control volume. Here the only body force of interest is the force of gravity or weight of the fluid.

(ii) Surface forces. These only act on the control surface; their effect on the fluid inside the control volume cancels out. They are always expressed in terms of stress (force per unit area). Two main types of surface force are involved namely.

(a) Pressure force. Pressure, p, is a stress that always acts perpendicular to the control surface and in the opposite direction to the unit normal (see Fig. 1.3). In other words it always tends to compress the fluid in the control volume. Although p can vary from point to point in the flow field it is invariant with direction at a particular point (in other words irrespective of the orientation of the infinitesimal control volume the

pressure force on any face will be —p6A where 6A is the area of the face) – see Fig. 1.3. As is evident from Bernoulli’s Eqn (2.16), the pressure depends on the flow speed.

(b) Viscous forces. In general the viscous force acts at an angle to any particular face of the infinitesimal control volume, so in general it will have two components in two-dimensional flow (three for three-dimensional flow) acting on each face (one due to a direct stress acting perpendicularly to the face and one shear stress (two for three-dimensional flow) acting tangentially to the face. As an example let us consider the stresses acting on two faces of a square infinitesimal control volume (Fig. 2.20). For the top face the unit normal would be j (unit vector in the у direction) and the unit tangential vector would be і (the unit vector in the x direction). In this case, then, the viscous force acting on this face and the side face would be given by

+ <Tyyj)6x X 1, {oxxl + СГхуІ)6у X 1

respectively. Note that, as in Section 2.4, we are assuming unit length in the z direction. The viscous shear stress is what is termed a second-order tensor – i. e. it is a quantity that is characterized by a magnitude and two directions (c. f. a vector or first-order tensor that is characterized by a magnitude and one direction). The stress tensor can be expressed in terms of four components (9 for three-dimensional flow) in matrix form as:

&XX &xy Gyx ®yy

Owing to symmetry axy = oyx. Just as the components of a vector change when the coordinate system is changed, so do the components of the stress tensor. In many engineering applications the direct viscous stresses (axx, oyy) are negligible compared with the shear stresses. The viscous stress is generated by fluid motion and cannot exist in a still fluid.

Other surface forces, e. g. surface tension, can be important in some engin­eering applications.

When the momentum equation is applied to an infinitesimal control volume (c. v.), it can be written in the form:

Rate of increase of momentum within the c. v.

‘————————- v———————– ‘

(i)

+ Net rate at which momentum leaves the c. v..

(И)

= Body force + pressure force + viscous force

V ^ s ^/ 41 s/ ^

(in) (iv) M

We will consider now the evaluation of each of terms (i) to (v) in turn for the case of two-dimensional incompressible flow.

Term (i) is dealt with in a similar way to Eqn (2.43), once it is recalled that momentum is (mass) x (velocity), so Term (i) is given by

To evaluate Term (ii) we will make use of Fig. 2.21 (c. f. Fig. 2.12). Note that the rate at which momentum crosses any face of the control volume is (rate at which mass crosses the face) x velocity. So if we denote the rate at which mass crosses a face by m, Term (ii) is given by

m3 x v3 – m, x у, + m4 x v4 – m2 x v2

But m3 and m are given by Eqns (2.38) and (2.39) respectively, and m2 and m4 by similar expressions. In a similar way it can be seen that, recalling v = (и, v)

/du 9v 6x [jhc’fa) T’
du <9v 6y ду’Ъу) T’
г64хУ4
/773 x $3
rhixv’i

So the x component of Eqn (2.61) becomes

du 8x

dv 6y dy 2

Cancelling like terms and neglecting higher-order terms simplifies this expression to

( du du dv

p[2ua-x + %+“^)lxSyx’

This can be rearranged as

f du du (du 9v)

/,[иГх + %+и^+а~И )SxSy x

= 0 Eqn (2.46)

In an exactly similar way the у component of Eqn (2.61) can be shown to be

( dv dv l’{ua-x+”Ty)6xlyxl

Term (iii) the body force, acting on the control volume, is simply given by the weight of the fluid, i. e. the mass of the fluid multiplied by the acceleration (vector) due to gravity. Thus

(2.63)

Normally, of course, gravity acts vertically downwards, so gx = 0 and gy = —g.

The evaluation of Term (iv), the net pressure force acting on the control volume is illustrated in Fig. 2.22. In the x direction the net pressure force is given by

6y x 1 = —^-SxSy x 1 (2.64a)

‘~lf)

(-fib*1

Similarly, the у component of the net pressure force is given by

dp,

dy

The evaluation of the x component of Term (v), the net viscous force, is illustrated in Fig. 2.23. In a similar way as for Eqn (2.64a, b), we obtain the net viscous force in the x and у directions respectively as

We now substitute Eqns (2.61) to (2.65) into Eqn (2.59) and cancel the common factor 8x8y x 1 to obtain

These are the momentum equations in the form of partial differential equations. For three dimensional flows the momentum equations can be written in the form:

where gx, gy, gz are the components of the acceleration g due to gravity, the body force per unit volume being given by pg.

The only approximation made to derive Eqns (2.66) and (2.67) is the continuum model, i. e. we ignore the fact that matter consists of myriad molecules and treat it as continuous. Although we have made use of the incompressible form of the continuity

бух 1

Eqn (2.46) to simplify Eqn (2.58a, b), Eqns (2.62) and (2.63) apply equally well to compressible flow. In order to show this to be true, it is necessary to allow density to vary in the derivation of Term (i) and to simplify it using the compressible form of the continuity Eqn (2.45).