BoundaryLayer Equations
The boundary layer can be considered as the phenomenological model of attached viscous flow. We derive in the following the boundarylayer equations for steady, compressible, threedimensional flow past a flat surface. We assume laminar flow, but note that the resulting equations also hold for turbulent flows, if we treat them as Reynolds – or Favreaveraged flows [1, 6].
The derivation is made in Cartesian coordinates. We keep the notation introduced in Fig. 4.1 with x and z (z not indicated there) being the coordinates tangential to the body surface, and y the coordinate normal to it.
Fig. 7.2. Streamline curvature and crossflow profiles of a threedimensional boundary layer (schematically) [3]: a) negative crossflow profile, b) sshaped crossflow profile, c) positive crossflow profile. The coordinates t and n are tangential to the body surface, г is normal to the body surface. 
Accordingly u and w are the tangential velocity components, and v is the component normal to the body surface.[86]
We try to keep the derivation as simple as possible in order to concentrate on the basic physical problems. (That is also the reason why we employ Cartesian coordinates.) Therefore, we do not include the description of the typical phenomena connected with hypersonic attached flow like the extra formulations for hightemperature realgas effects, surfaceradiation cooling, and slipflow effects. Their introduction into the resulting system of equations is straight forward.
The boundarylayer equations are derived from the NavierStokes equations, together with the continuity equation and the energy equation, SubSection 4.3. They cannot be derived from first principles [1]. One has to introduce the observation—the boundarylayer assumption, originally conceived by L. Prandtl [7]—that the extension of the boundary layer in direction normal to the body surface (coordinate y and the involved boundarylayer thicknesses) is very small, like also the velocity component v in the boundary layer normal to the body surface. Actually the observation is that the different boundarylayer thicknesses and v in ydirection are inversely proportional to the square root of the Reynolds number.
We take care of this observation by introducing the socalled boundary – layer stretching, which brings y and v, nondimensionalized with reference data Lref and vref, respectively, to O(1):
Reref = Preft’re/Lre/ . (7.3)
ftref
The prime above denotes variables, which were nondimensionalized and stretched, however, we use it in the following also for variables, which are only nondimensionalized.
All other variables are simply made dimensionless with appropriate reference data, and assumed then to be O(1): velocity components u and w with vref, lengths x and z with Lref, temperature T with Tref, density p with pref, pressure p with prefv? ff, the transport coefficients /л and к with /i, ref and kref, respectively, and finally the specific heat at constant pressure cp with cPref.4 Each resulting dimensionless variable is marked by a prime, for instance
u’ = —. (7.4)
vref
We introduce boundarylayer stretching and nondimensionalization first into the continuity equation, eq. (4.83). We do this for illustration in full detail. In three dimensions and without the partial time derivative, Section 4.1, we replace и with u’vref, eq. (7.4), v with v’vrefjjReref, eq. (7.2), etc, and find
df/prefu’vref ^ dp’prefv’yref/sjReref ^ dp’pref w’vref _ 0 ^ ^
Ox Lref dy’Lref/ уReref k)z Lrej
Because all reference parameters, and also Reref are constants, we find immediately the stretched and dimensionless continuity equation which has the same form—this does not hold for the other equations—as the original equation:
4 In classical boundarylayer theory the pressure is made dimensionless with Pref v2ref, which has the advantage that the equations describe in this form both compressible and incompressible flows. For general hypersonic viscous flows, however, we choose pref to make the pressure dimensionless, SubSection 7.1.7.
Again we introduce nondimensional and stretched variables, as we did above. We also write explicitly all terms of O(1), and bundle together all terms, which are of smaller order of magnitude, now except for two of the heatconduction terms:and Eref the reference Eckert number:
Eref = (jref — l)Mref ■ (7.22)
(Both were introduced in SubSection 4.3.2.)
We have retained on purpose in this equation two terms, which are nominally of lower order of magnitude. They are the gradients of the heat – conduction terms in x and z direction, which are of O(l/Reref). The reason is that we in general consider radiationcooled surfaces, where we have to take into account possible strong gradients of Tw in both x and z direction. They appear there on the one hand, because the thermal state of the surface changes strongly in the downstream direction, usually the mainaxis direction of a flight vehicle, Chapter 3. On the other hand, strong changes are present in both x and z direction, if laminarturbulent transition occurs, see, e. g., SubSection 7.3.
The question now is, under what conditions can we drop the two terms, regarding the changes of the wall temperature. To answer it, we follow an argumentation given by Chapman and Rubesin [10]. We consider first (in two dimensions) the gradient of the heatconduction term in direction normal to the surface in dimensional and nonstretched form, eq. (7.19), and introduce finite differences, as we did in SubSection 3.2.1:
with 5T being the thickness of the thermal boundary layer.
The gradient of the heatconduction term in xdirection is written, assuming that (dT/dx)w is representative for it
The result is: provided, that eq. (7.26) holds, the gradient term of heat conduction in жdirection can be neglected, because St/L ос 1/(/Pr/Re) ^ 1 in general means
d dT d dT
« Ж,(к 17’■
In [10] it is assumed, that the recovery temperature is representative for the wall temperature
Tw=Tr= TU1 + г^ІАф. (7.29)
Introducing this into eq. (7.26), together with St ~ S « cL/^/Reref for laminar flow, we obtain
With r = y/Pr = a/0.72, Y = 1.4, c = 6 we arrive finally, after having introduced nondimensional variables, at the ChapmanRubesin criterion [10]. It says that the term eq. (7.24) can be neglected, if
dT’
— ш ^ <U>3.l/:( x H,,. (7.31)
This means, that, for instance, for M^ = 1 and Reref = 106, the maximum permissible temperature gradient would be equivalent to a thirtyfold increase of T/T^ along a surface of length Lref. From this it can be concluded, that in general for high Machnumber and Reynoldsnumber flows the ChapmanRubesin criterion is fulfilled, as long as the surfacetemperature distribution is “reasonably smooth and continuous”. The situation can be different for low Mach numbers and Reynolds numbers.
With radiationcooled surfaces, as we noted above, we do not necessarily have reasonably smooth and continuous surfacetemperature distributions in both x and z direction. Moreover, the basic relation eq. (7.23) needs to be adapted, because it does not describe the situation at a radiationcooled surface. For that situation we introduce a slightly different formulation for both directions:
(7.32)
(7.32) because, at least for laminar flow, Tr — Tw is the characteristic temperature difference, see SubSection 3.2.2. We also introduce the absolute values dT/dxw and dT/dzw, because the gradients will be negative downstream
of the forward stagnation point, SubSection 3.2.1, but may be positive or negative in laminarturbulent transition regimes, SubSection 7.3, and in hotspot and coldspot situations, SubSection 3.2.4.
The modified ChapmanRubesin criterion is then: if eqs. (7.32) and (7.33) hold, the gradient term of heat conduction in both x and ^direction can be neglected, because again St/L ос 1/(/Pr/Re)
__L___ d_
Reref dx’
1 d ‘ Reref dz
We refrain to propose detailed criteria, like the original ChapmanRubesin criterion, eq. (7.31). This could be done for the region downstream of the forward stagnation point, but not in the other regimes. In practice the results of an exploration solution should show, if and where the modified Chapman Rubesin criterion is violated or not and whether the two tangential heat conduction terms must be kept or not.[87]
Provided that the modified ChapmanRubesin criterion is fulfilled, we arrive at the classical boundarylayer equations by neglecting all terms of O(1/Reref) and O(1/Re2ref) in eqs. (7.6), (7.16) to (7.18), and (7.20). We write the variables without prime, understanding that the equations can be read in either way, nondimensional, stretched or nonstretched, and dimensional and nonstretched, then without the similarity parameters Prref and
Eref :
dpu dpv dx dy 
g о c§" ^ + 
= 0, 
(7.36) 

du 
du 
du 
dp 
d 
f du 
(7.37) 
fmd~x 
+ ,n% + 
PWTz 
dx 
+ % 1 
V%) ’ 

0 = – 
dp dy’ 
(7.38) 

dw 
dw 
dw 
dp 
d 
f dw 
(7.39) 
PU^ 
+ ""% + 
pwlb 
dz 
+ dy 
Ы) ’ 
du 2 / dw 2 1 dy) +dy) j 
These equations are the ordinary boundarylayer equations which describe attached viscous flow fields on hypersonic flight vehicles. If thick boundary layers are present and/or entropylayer swallowing occurs, they must be employed in secondorder formulation, see below. For very large reference Mach numbers Mref the equations become fundamentally changed, see SubSection 7.1.7.[88]
With the above equations we can determine the unknowns u, v, w, and T. The unknowns density p, viscosity p, thermal conductivity k, and specific heat at constant pressure cp are to be found with the equation of state p = pRT, and the respective relations given in Chapters 4 and 5. If the boundary – layer flow is turbulent, the apparent transport properties must be introduced, [1] and Section 8.5. If hightemperature realgas effects are present in the flow field under consideration, the respective formulations and laws must be incorporated.
Since dp/dy is zero, the pressure field of the external inviscid flow field, represented by dp/dx and dp/dz, is imposed on the boundary layer. This means, that in the boundary layer dp/dx and dp/dz are constant in y – direction. This holds for firstorder boundary layers. If secondorder effects are present, dp/dx and dp/dz in the boundary layer are implicitly corrected via dp/dy = 0 by centrifugal terms, see below.
The equations are firstorder boundarylayer equations, based on Cartesian coordinates. In general locally monoclinic surfaceoriented coordinates, factors and additional terms are added, which bring in the metric properties of the coordinate system [1]. It should be noted, that the equations for the general coordinates are formulated such that also the velocity components are transformed. This is in contrast to modern Euler and Navier – Stokes/RANS methods formulated for general coordinates. There only the geometry is transformed, Section A, and not the velocity components.
If locally the boundarylayer thickness is not small compared to the smallest radius of curvature of the surface, the pressure gradient in the boundary layer in direction normal to the surface, dp/dy, is no longer small of higher order, and hence no longer can be neglected.[89] This is a situation found typically in hypersonic flows, where also entropylayer swallowing can oc
cur, SubSection 6.4.2. This situation is taken into account by secondorder boundarylayer equations, which basically have the same form as the first – order equations [1, 12]. Information about the curvature properties of the surface is added. The ymomentum equation does not degenerate into dp/dy = 0. dp/dy is finite because centrifugal forces have to be taken into account. At the outer edge of the boundary layer the boundary conditions are determined by values from within the inviscid flow field, not from the surface as in firstorder theory, see the discussion in SubSection 6.4.2. Also the first derivatives of the tangential velocity components, of temperature, density and pressure are continuous [12], which is not the case in firstorder theory, see, e. g., Fig. 6.23.
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