1.7 The Navier–Stokes Equations
The Navier–Stokes equations are obtained by using the constitutive equation (1.5.25) in the momentum (1.6.27) and energy (1.6.38) equations. At this stage, it is convenient to consider incompressible and compressible fluid flows separately.
1.7.1 Incompressible fluid flows
In an incompressible flow, the fluid density ρ is a constant quantity, which means that the continuity equation (1.6.6) may be written as
divV = 0. (1.7.1)
Consequently, the constitutive equation (1.5.25) reduces to
pij =−pδij+ µ
∂Vi
∂xj +∂Vj
∂xi
. (1.7.2)
As has already been mentioned in Section 1.5, liquid flows, as well as slow gas flows, are incapable of producing a noticeable variation of temperature. Therefore, the viscosity coefficient µ may be assumed constant in (1.7.2).
Now, we need to write equations (1.6.26) using index notations. We see that, with i playing the role of the summations index, the projection of divP upon the xj-axis is
divP
j= ∂pij
∂xi
, j = 1, 2, 3. (1.7.3)
Substitution of (1.7.2) into (1.7.3) yields
divP
j=−∂p
∂xj
+ µ ∂
∂xi
∂Vi
∂xj
+ µ ∂2Vj
∂xi∂xi
.
Changing the order of differentiation in the second term,
∂
∂xi
∂Vi
∂xj
= ∂
∂xj
∂Vi
∂xi
= ∂
∂xj
divV ,
and using the continuity equation (1.7.1), we arrive at the conclusion that this term is zero. Hence,
divP
j =−∂p
∂xj + µ ∂2Vj
∂xi∂xi, j = 1, 2, 3.
In vector form, this equation is written as
divP = −∇p + µ∇2V, (1.7.4)
where the symbol∇2Vdenotes a vector whose components are ∇2u,∇2v,∇2w .
With (1.7.4), the momentum equation (1.6.27) becomes13 DV
Dt = f−1
ρ∇p + ν∇2V. (1.7.5a)
This should be supplemented with the continuity equation
divV = 0 (1.7.5b)
to form a closed set of Navier–Stokes equations that govern the motion of incompress-ible fluids. In coordinate decomposition form, they are written as
∂u
These four equations involve four unknown functions: the velocity components u, v, w and the pressure p. It should be noted here that when calculating the velocity and pressure fields in an incompressible flow, we do not need the energy equation.
1.7.2 Compressible fluid flows
We start again with the momentum equation (1.6.27). This may be projected on the three coordinate axes. The x-component of the momentum equation is written, using the first of equations (1.6.26), as
ρDu
Dt = ρfx+∂pxx
∂x +∂pyx
∂y +∂pzx
∂z . (1.7.7)
From the constitutive equation (1.5.25), it is easily found that pxx=−p + µ
13Recall that the kinetic viscosity ν = µ/ρ.
1.7. The Navier–Stokes Equations 63 which, after substitution into (1.7.7), yields
ρ Similarly, the y-component of the momentum equation may be shown to be
ρ and the z-component is written as
ρ Let us now turn to the energy equation (1.6.38),
ρDe
Dt = div(PV) − V · div P + div(κ∇T ). (1.7.11) Remember that PV coincides with the vector A whose components are given by equation (1.6.35). Consequently,
Taking into account that the components of the vector divP are given by (1.7.3), we can easily see that
Vj
∂pij
∂xi
= V· div P. (1.7.13)
Substituting (1.7.13) into (1.7.12) and then into (1.7.11) renders the energy equation in the form
ρDe
Dt = pij∂Vj
∂xi
+ div(κ∇T ). (1.7.14)
Using the constitutive equation (1.5.29), we can express the first term on the right-hand side of (1.7.14) as
pij
Direct summation with respect to indices i, j = 1, 2, 3 yields
∂Vj and we can conclude that the energy equation (1.7.14) may be written in the form
ρDe For an incompressible fluid flow, equation (1.7.15) turns into the following equation for the temperature T :
ρcv Indeed, the first and second terms on the right-hand side of (1.7.15) disappear in view of the fact that in incompressible flows divV = 0. When dealing with the internal energy e on the left-hand side of (1.7.15), equation (1.3.26) has been used. Also, we have used the fact that in incompressible flows, the heat transfer coefficient κ and the viscosity coefficient µ are constant throughout the flow field. Equation (1.7.16) allows us to calculate the temperature variations over the flow field after the velocity field has been found by solving equations (1.7.6).
Now, returning to compressible flows, it is convenient to express the energy equation (1.7.15) as an equation for the enthalpy h. In order to perform this task, we start with the continuity equation (1.6.6):
∂ρ
∂t + div(ρV) = 0. (1.7.17)
1.7. The Navier–Stokes Equations 65
Hence, the continuity equation (1.7.17) may be written as Dρ
Dt + ρ divV = 0,
and therefore, the first term on the right-hand side of (1.7.15) can be substituted by
−p divV = p ρ
Dρ Dt. Using further the identity
p
and taking into account that the enthalpy is defined as h = e + p/ρ, we see that the energy equation (1.7.15) may be written in the form
ρDh In this book series, our main attention will be on perfect gas flows, in which case the enthalpy h is given by equation (1.3.28),
h = cpT.
Therefore, assuming cp constant, we have
∇T = 1 cp∇h.
This equation can be rearranged, using (1.6.32), as κ∇T = µ
P r∇h. (1.7.19)
It remains to substitute (1.7.19) into (1.7.18), and we have the energy equation for a perfect gas flow in the form
ρDh
Collecting together the momentum equations (1.7.8), (1.7.9), (1.7.10), the energy equation (1.7.20), the continuity equation (1.6.6), and the perfect gas state equation (1.3.33), we can write the Navier–Stokes equations governing the motion of a perfect gas as
1.7. The Navier–Stokes Equations 67 1.7.3 Integral momentum equation
If the solution of the Navier–Stokes equations is found for a particular flow, say, for a flow past a rigid body, then the total force experienced by the body may be calcu-lated through integration of the stress pn produced by the moving fluid on the body surface. However, it often happens that only partial information about the flow field is available. For example, an experimentalist might want to avoid disturbing the flow in the immediate vicinity of the body surface, and choose to perform the measurements at some distance from the body. A theoretician might be able to find the behaviour of the solution in the ‘far field’, but not close to the body. The force experienced by the body may still be calculated, for which purpose the integral momentum equation is used.
When deducing this equation, we shall assume that the flow considered is steady (∂V/∂t = 0) and the body force is negligibly small (f = 0). Then the momentum equation (1.6.27) takes the form
ρ V· ∇
V= divP.
Projecting this equation upon the xα-axis, we have ρVi
∂Vα
∂xi
= ∂piα
∂xi
, α = 1, 2, 3. (1.7.22)
Here the xα-component of divP is calculated according to the rule given by equations (1.6.26).
For a steady flow, compressible or incompressible, the continuity equation (1.6.6) is written as
div(ρV) = 0, or, equivalently,
∂
∂xi
ρVi
= 0. (1.7.23)
Multiplying (1.7.23) by Vα and adding the result to (1.7.22) yields
∂
∂xi
ρViVα
=∂piα
∂xi
. (1.7.24)
Hence, if we introduce a vector A whose components are defined as
Ai= ρViVα− piα, i = 1, 2, 3, (1.7.25) then we can write equation (1.7.24) as
divA = 0.
Let us integrate this equation over the region D, termed control volume, that lies between the body surface Sband an arbitrarily chosen surface Sc embracing the body
n
n
Sb
Sc
D
Fig. 1.28: Control volumeD.
as shown in Figure 1.28; both the body and control surface are assumed to be fixed with respect to an inertial coordinate system (x1, x2, x3). We have
ZZZ
D
divA dτ = 0.
Then using Gauss’s divergence theorem, we can write ZZ
Sc
A· n ds +
ZZ
Sb
A· n
ds = 0, (1.7.26)
where n is the unit normal vector external toD.
It follows from (1.7.25) and (1.2.11) that
A· n = Aini= ρVα(Vini)− piαni= ρVα V· n
− pnα, which, when substituted into (1.7.26), yields
ZZ
Sc
h
ρVα V· n
− pnαi ds +
ZZ
Sb
h
ρVα V· n
− pnαi ds = 0.
Since this equation is valid for α = 1, 2, 3, it may be written in a vector form ZZ
Sc
hρ V· n
V− pni ds +
ZZ
Sb
hρ V· n
V− pni
ds = 0. (1.7.27)
Let us examine the second integral in (1.7.27). We note, first, that the surface of a rigid body is impenetrable by a fluid. This means that the normal velocity component Vn = V· n is zero everywhere on Sb. Second, returning to (1.2.3), we recall that pnds is the force acting through ds on the fluid situated on the rear side of Sb. With
1.7. The Navier–Stokes Equations 69 the direction of n as shown in Figure 1.28, this is the force experienced by the fluid surrounding the body. Hence, the force acting upon the body is given by
F=− Z Z
Sb
pnds. (1.7.28)
Combining (1.7.27) and (1.7.28), we have F=
ZZ
Sc
pnds− ZZ
Sc
ρ V· n
Vds. (1.7.29)
This shows that the force exerted by the flow on a rigid body equals the force acting on the control surface Sc minus the momentum flux through this surface.
Of course, if there is no rigid body inside the control surface Sc and the entire regionD is filled with moving fluid, then equation (1.7.29) reduces to
ZZ
Sc
ρ V· n Vds =
ZZ
Sc
pnds, (1.7.30)
stating that the momentum flux through a control surface equals the total surface force acting on this surface.
1.7.4 Similarity rules in fluid dynamics
When solving the Navier–Stokes equations (1.7.21), one needs to use an appropriate set of boundary conditions. The form of the latter depends on the particular problem considered. Here we shall assume that we are dealing with a rigid body, say, an aircraft wing, placed in a uniform flow of a perfect gas. We shall denote the gas velocity far from the body (termed the free-stream velocity) as V∞, the free-stream pressure as p∞, the gas density in the free stream as ρ∞, and the dynamic viscosity coefficient as µ∞. The coordinate system can always be rotated to make the x-axis parallel to the free-stream velocity vector, and then we will have
u = V∞, v = w = 0, p = p∞, ρ = ρ∞ at x2+ y2+ z2=∞. (1.7.31) Let us further assume that the body force f is that due to gravity. For an aircraft in cruise, this is directed vertically downwards, i.e.
fx= fz= 0, fy =−g, (1.7.32)
where g is the acceleration of free fall.
Finally, the boundary conditions on the wing surface S have to be formulated. We shall assume that the wing surface equation may be written in the form
Φ
x
L,y− y0(t) L , z
L
= 0. (1.7.33)
Here the arguments of the function Φ are made dimensionless using the wing chord L.
This allows us to consider a family of wings that are geometrically similar to one another but have different size L.
Equation (1.7.33) also allows for wing oscillations to be included in the discussion, as we want to model phenomena such as wing flutter.14For our purposes, it is sufficient to assume that the wing moves as a whole along the y-axis, being described by the equation
y0(t) = L¯y0
t T
,
with the period T of the oscillation used to make the argument of the function ¯y0
dimensionless.
The fluid velocity has to satisfy the following conditions on the body surface S:
u = w = 0, v = dy0
dt on S. (1.7.34)
These conditions represent a fundamental property of fluid motion, which consists in the following observation. When a fluid particle comes in contact with a solid body, it always assumes the same velocity as the corresponding element of the body surface.
This is attributed to the molecular forces acting between fluids and solids. They prevent the fluid from ‘sliding’ along the body surface. The resulting restriction on the fluid velocity field is called the no-slip condition. Theoretical justification of this result is still to be found for a general fluid flow, and therefore one has to rely on the available experimental evidence. The latter is ample and supports the view that the no-slip condition is a universal law of fluid dynamics.
In addition to (1.7.34), one has to formulate a boundary condition for the en-thalpy h. If we assume that the body surface S is thermally isolated, i.e. there is no heat transfer through S, then
∇h · n = 0 on S. (1.7.35)
This closes the formulation of the boundary-value problem for the Navier–Stokes equa-tions (1.7.21).
We shall now cast the equations (1.7.21) and the boundary conditions (1.7.31), (1.7.34), (1.7.35) in non-dimensional form. We introduce the non-dimensional variables (denoted by the ‘bar’) using the following transformations
t = T ¯t, x = L¯x, y = L¯y, z = L¯z, u = V∞u,¯ v = V∞¯v, w = V∞w,¯
p = p∞+ ρ∞V∞2p,¯ ρ = ρ∞ρ,¯ h = V∞2¯h, µ = µ∞µ.¯
(1.7.36)
Substitution of (1.7.36) into (1.7.21) results in
14Flutter is observed when an aircraft’s speed exceeds a critical value, and then violent oscillations of the wing take place, which can result in destruction of the wing frame.
1.7. The Navier–Stokes Equations 71
The free-stream boundary conditions (1.7.31) are written in the non-dimensional vari-ables as
¯
u = ¯ρ = 1, ¯v = ¯w = ¯p = 0 at x¯2+ ¯y2+ ¯z2=∞, (1.7.38a) and the conditions on the wing surface, (1.7.34) and (1.7.35), assume the form
¯
u = ¯w = 0, v = St¯ d¯y0
d¯t , ∇¯h · n = 0 on S. (1.7.38b)
The equations (1.7.37) and boundary conditions (1.7.38) involve six non-dimensional parameters. Two of these, the ratio of specific heats γ and the Prandtl number P r, were introduced in Sections 1.3.2 and 1.6.3, respectively. The four ‘new’ parameters are
Reynolds number Re = ρ∞V∞L µ∞ ,
Mach number M∞= V∞
pγp∞/ρ∞
,
Strouhal number St = L V∞T, Froude number F r = V∞
√gL.
Provided that the solution of the boundary-value problem (1.7.37), (1.7.38) exists and is unique, the following statement is valid. If two wings with different chords L and different periods of oscillations T are placed into two flows with different V∞, p∞, and ρ∞, but (i) the functions Φ and ¯y0 representing the wings’ geometry and motion are the same and (ii) the similarity parameters Re, M∞, St, F r, P r, and γ are also the same in the two flows, then these flows will appear identical in the non-dimensional variables. The first of the above conditions is known as geometric similarity and the second as dynamic similarity.
The similarity properties of fluid flows are widely used by experimentalists. In particular, an essential part of the aircraft design process is wind tunnel testing. These tests are performed on scaled-down models of the entire aircraft or its elements. To achieve dynamic similarity of the flow in a wind tunnel with that in real flight, one needs to reproduce the Mach number M∞and the Reynolds number Re as closely as possible,15 and then the velocity, density and pressure fields are expected to be the same in the non-dimensional variables. The dimensional quantities are recalculated in an obvious way, keeping in mind that
u V∞, v
V∞, w V∞, ρ
ρ∞, p− p∞ ρ∞V∞2 remain the same at the corresponding points in the flow field.
Exercises 4
1. Does the continuity equation
∂ρ
∂t + div(ρV) = 0
admit a simplification in the case when ρ remains unchanged for each fluid particle but is different for different fluid particles? An example of such a fluid is sea water.
It has different salt concentration, depending on the water depth.
15In aerodynamic flows, one does not need to think of the Froude number, since gravitational effects are too small. Also, when dealing with a steady flow, typical of cruise flight, the Strouhal number becomes unimportant.