.. 

NASA Contractor Report 178268 

I 

ICASE REPORT NO. 87-16 

ICASE .. THE FULLY NONLINEAR DEVELOPMENT OF GORTLER 

VORTICES IN GROWING BOUNDARY LAYERS 

\ ~_ 

16ASA-CR- 178266) TEE EULLY NCLLILEAB 1587-20502 

GEVELOPWEHT OE GCEBTLEE VCii'IICES Ih: GROWING EGONEBBY LAYEPS (LASA) 36 f cscz 20D 

Philip Hall 

William D. Lakin 

Contract No. NAS1-18107 

March 1987 

INSTITUTE FOR COMPUTER APPLICATIONS IN SCIENCE AND ENGINEERING 

NASA Langley Research Center, Hampton, Virginia 

Operated by the Universities Space Research Association 

23665 

NationalAeronauticsand 

Space Administration 

Hampton,Mrginia 23665 

m@@yRamch- 
The fully nonlinear development of Gortler vortices in growing boundary layers 

by Philip Hall, Mathematics Department, University of Exeter 

and William D. Lakin, Mathematics Department, Old Dominion University. 

Abstract 

The fully nonlinear development of small wavelength Giirtler vortices in a growing 

boundary layer is investigated using a combination of asymptotic and numerical methods. 

The starting point for the analysis is the weakly nonlinear theory of Hall (1982b) who 

discussed the initial development of small amplitude vortices in a neighbourhood of the 

location where they first become linearly unstable. That development is unusual in the 

context of nonlinear stability theory in that it is not described by the Stuart-Watson 

approach. In fact the development is governed by a pair of coupled nonlinear partial 

differential evolution equations for the vortex flow and the mean flow correction. Here the 

further development of this interaction is considered for vortices so large that the mean flow 

correction driven by them is as large as the basic state. Surprisingly it is found that such 

a nonlinear interaction can still be described by asymptotic means. It is shown that the 

vortices spread out across the boundary layer and effectively drive the boundary layer. In 

fact the system obtained by writing down the equations for the fundamental component of 

the vortex generate a differential equation for the basic state. Thus the mean flow adjusts 

so as to make these large amplitude vortices locally neutral. Moreover in the region where 

the vortices exist the mean flow has a ‘square-root’ profile and the vortex velocity field 

can be written down in closed form. The upper and lower boundaries of the region of 

vortex activity are determined by a free-boundary problem involving the boundary layer 

equations. In general it is found that this region untimately includes almost all of the 

original boundary layer and much of the free-stream. In this situation the mean flow has 

essentially no relationship to the flow which exists in the absence of the vortices. 

, 

This work was supported under the National Aeronautics and Space Administration 

under NASA Contract No. NAS1-18107 while the authors were in residence at ICASE, 

NASA Langley Research Center ,Hampton ,VA 23665. 

i 
Introduction 

Our concern is with large amplitude Giirtler vortices in viscous incompressible flows 

over walls of variable curvature. Much of the recent interest in the Gortler instability 

has been motivated by practical problems such as Laminar Flow Control or the flow over 

turbine blades. A particular cause for concern in Laminar Flow Control is the question 

of whether Gortler vortices are likely to induce premature transition because of their 

effect on the receptivity of the original boundary layer to Tollmien-Schlichting waves or 

crossflow vortices. As yet little progress has been made with the latter problem but 

the corresponding internal fully developed flow problem has been discussed by Hall and 

Bennett (1986), Bennett and Hall, (1987) and Hall and Smith (1987). Thus, for example, 

Bennett and Hall (1987) showed that fully,,nonlinear Taylor vortices in a curved channel 

can have a massive influence on the growth of lower branch Tollmien-Schlichting waves. 

The absence of a fully nonlinear description of Gijrtler vortices in external flows means that 

a similar interaction problem cannot yet be attempted for these flows. Here we shall show 

how the required large amplitude vortex flows can be described asymptotically. However 

it is perhaps useful if we first describe earlier work on the Gortler problem. 

The original work of Gortler (1940) and the later work of, for example, Hammerlin 

(1955, 1956), Smith (1955) and Floryan and Saric (1979) did not take care of boundary 

layer growth in a selfconsistent manner. Unlike the corresponding Tollmien-Schlichting 

instability problem, where the fact that instability occurs at relatively high Reynolds 

numbers renders parallel flow calculations reasonably accurate, the neglect of non-parallel 

effects in the Gortler problem leads to inconsistent and, in some cases, physically absurd 

results. Thus as extreme examples some of the results predicted by these calculations 

showed instability at zero Gortler number or zero wavenumber. 

In fact at high wavenumbers the above theories gave consistent results and Hall (1982a) 

showed that in this regime an asymptotic solution of the non-parallel problem is possible. 

The vortices are found to become linearly unstable at a particular downstream location and 

to concentrate themselves at some depth in the boundary layer. This depth corresponds to 

the position where Rayleigh’s inviscid instability criterion for the boundary layer is most 

violated. At O( 1) wavenumbers no asymptotic or self-consistent parallel flow calculation is 

possible and the linear instability partial differential equations must be solved numerically 

as was done by Hall (1983). These equations are parabolic in the downstream variable 

and so can be solved by a marching procedure and the neutral location for a disturbance 

inserted at a given position can be calculated. Hall (1983) showed that the neutral curve 

produced by such a calculation depends on the nature of the initial disturbance. However 

each such neutral curve has none of the obvious anomalies of the parallel flow neutral 

curves and at high wavenumbers the different neutral curves merge into the asymptotic 

result . 

The initial nonlinear development of Gijrtler vortices of small waveiength was discussed 

by Hall (1982b). It was found that the nonlinear interaction is dominated by a ‘mean-field’ 

type theory in which the fundamental mode and mean flow correction reinforce each other. 

The higher harmonics play no role in this interaction so the Stuart-Watson approach is 
2 

not applicable. In order to motivate the asymptotics to be used for much larger vortices 

it is instructive to summarize the essential details of the calculation of Hall (1982b) 

Suppose then that a vortex of nondimensional wavenumber E-' is locally neutrally 

stable at the downstream location x = z of a two-dimensional boundary layer flow (ti, D). 

If the most unstable part of the boundary layer is at a depth y = g it is necessary to define 

variables X and by 

X= 9 (l.lu, b) e=- (x - 2) Y-Q 

E ET 

and the total downstream velocity component expands in the neighbourhood of (2, Q) as 

For a boundary layer which has a local Gortler number increasing faster than the fourth 

power of the local wavenumber the evolution equation for uM, the mean flow correction, 

and the fundamental U1 take the form 

aUM -e + x u1 = 2u1---. {ai.-m-4 1 ac 

a2 2a 1, (1.3u, b) 

The linearized form of (1.3) shows that U1 can be expressed in terms of parabolic cylinder 

functions. Otherwise (1.3) must be solved numerically by a marching procedure. However 

at large values of X it was shown by Hall (1982b) that UM and U1 develop a surprisingly 

simple asymptotic structure. In fact UM and U1 can be written down explicitly in terms of 

X and a similarly variable € in a finite interval €1 < 6 < €2. Near €1, €2 the fundamental 

satisfies a nonlinear Airy equation and is reduced to zero exponentially whilst UM persists 

above €2 and below El. Of crucial importance is the fact that when this structure develops 

UM - X3/2, U1 - X1/, so that the total mean flow correction c3j2u~ is comparable with 

the basic state for X - I/E. Thus if x - z is O(1) the vortices will drive a mean flow 

correction as big as the original mean state. 

The remarkable feature of the latter regime is that the nonlinear Gortler equations can 

still be solved asymptotically in this region. Thus we are able to describe asymptotically 

a large amplitude disturbance capable of altering at zeroth order the basic state. The 

structure of the large X solution of Hall (1982b) continues to describe the flow but with 

the major change that the depth of the fluid where vortex activity persists is now O(1). In 

this layer the mean flow is driven by the vortices and indeed is determined as a solvability 

condition on the equations for the fundamental there. The downstream mean velocity 

component in this layer then has a simple square root profile and the mean equations 

drive a finite amplitude vortex. Thus there is an exact reversal of the usual roles of 

the equations for the mean and fundamental obtained by Fourier- analysing the spanwise 
3 

c2pendence o the disturbance. The situation is not unlike the scenario postulated some 

years ago for turbulent flows by Malkus (1956). We recall that Malkus argued that the 

‘mean’ part of a turbulent flow would organize itself so that any ‘modes’ were marginally 

stable. We shall refer to the layer where the vortex activity is concentrated as I. This layer 

is bounded by ‘transition’ layers of depth c2/3 where the vortices are again reduced to zero 

as solutions of a nonlinear Airy equation. We refer to these layers as IIa,b respectively 

and the remainder of the flow is denoted by regions IIIa, b as shown in Figure 1. 

In regions IIIa, b there is no spanwise dependence for the flow and the velocity field 

satisfies the boundary layer equations. The solutions of these equations must be matched 

with those emanating from IIa, b whose po2itions are unknown functions of the downstream 

variables. Thus a complete description of the flow requires the numerical solution of a freeboundary partial differential system. 

The region I of vortex activity in general grows as the flow moves downstream until 

it eventually occupies almost all of the extent of the boundary layer which would exist in 

the absence of the vortices. In fact the only part free of vortices is a thin layer at the 

wall and the vortices even take over part of the free stream. At this stage the flow has 

effectively no relationship with the flow which would exist in the absence of the vortices. 

The procedure adopted in the rest of this paper is as follows: in $2 the nonlinear Gijrtler 

vortex equations are derived. In $3 an asymptotic solution of these equations for large 

amplitude small wavelength vortices is given. In $4 an asymptotic solution of the initial 

and ultimate downstream stages of the free-boundary problem obtained in $3 is discussed. 

In $5 a numerical scheme which we have used to solve this free boundary problem is 

described. Finally in 56 we discuss our results and draw some conclusions. 

2. Formulation of the nonlinear Gortler equations. 

We consider the flow of a viscous fluid of kinematic viscosity v, density p, over a wall 

of variable concave curvature a-’x(X/L). Here X denotes distance along the wall, a is a 

typical radius of curvature and L is a typical length scale along the wall. If UO is a typical 

flow velocity we define a Reynolds number RE by 

UOL RE = -, 

U 

and a curvature parameter 6 by 

L 

a 

6 = -. 

We confine our attention to the limit RE + 00 with the Giirtler number G defined by 

held fixed. We take (X, Y, 2) to be co-ordinates along the wall, normal to the wall and in 

the spanwise direction respectively. If (U, V, W) denotes the corresponding velocity vector 

we define dimensionless co-ordinates (z, y, z) and velocity (u, v, w) by 

1 

(5, y, z) = L-’(X,YRi, ZRj) 
4 

and 

(U,V,W,) = Uo(u,vR;1/2,WR;1/2). 

We restrict our analysis to flows with u 3 1, y --+ 00 so that the pressure P can be written 

in the form 

uo2 P = p-p 

RE (2.4) 

The Navier-Stokes equations for the flow can be written in the form 

du av aw - + - + - = 0, 

ax dy az 

du du au 

vu=u-+v- +w-, 

ax dy az 

1 2 dp dv dv dV 

2 dy ax ay az 

VU - -GXU - - = U- + V- + W-, 

dp dw aw aW 

vw - - = u-+v-+ w-, dz ax dy az 

(2.5a, b, c, d) 

d2 d2 - 1/2 where V = - + - and terms of relative order RE have been neglected. We further 

du2 dz2 

note that wehave assumed that the flow is steady. 

In the absence of any Gortler vortex we can write (u, v, w) = (a, 8,O) in which case 

a,V satisfy 

a = jj = 0, y = 0, 5 + 1, v + 00, 5 = 1,s = 0,y > 0. 

Thus (ti 0 0) is just a Blasius boundary layer until a Gijrtler vortex begins to grow at some 

downstream location 

Finally in this section we note that our analysis can easily be modified to take care of 

pressure gradient driven flows. In that case P in (2.4) must be altered so as to contain an x 

dependent component of size pU: and the boundary conditions in (2.6) must be modified. 

We shall obtain a solution of (2.5) which satisfies 

(2.7a, b) 

In fact we shall see that when y + 00,v + v(x),w --+ 0; in general v(x) is not the 

y velocity component at infinity of a Blasius boundary layer so the higher order outer 

‘inviscid’ problem associated with (2.5) is modified by the instability. 
5 

3. The evolution equations for large amplitude Gortler vortices 

We now develop an asymptotic solution of (2.5) valid in the limit of small vortex 

wavelength. We suppose that the boundary layer (a(x, y), ~(x, y)) becomes linearly unstable to Gortler vortices at x = x*. Furthermore we assume that the curvature function 

x(x) is such that the flow becomes more unstable with increasing x. Thus if the boundary 

layer is a Blasius boundary layer we require that x increases more rapidly than xh. This 

restriction is discussed by Hall (1983) and is a direct consequence of the scaling of the right 

hand branch of the neutral curve for Giirtler vortices. 

The discussion of $2 suggests that beyond x = x* the flow will support fully nonlinear 

Gortler vortices. The flowfield is therefdre split up as shown in Figure 1. The vortex 

activity is therefore confined between yl(&) and y2(x). The layers denoted by regions 

IIa, b are required in order to smooth out the algebraically decaying vortices in region I. 

Later we will see that these layers must be of thickness in the limit E --+ 0. We expand 

the Gortler number G in the form 

whilst in region I the appropriate expansions of u,v,w and p are 

u = a, + Eli1 + * * + {€E(u; + EU; + * * -) + €2E2(UZ + EU? + * - .) * - - + C.C.} 

21 = Bo + €81 + * * - + {e'E(V,1 + €v; + - .) + E E (V,2 + EV? + * * .) * * * + C.C.} 

w = E(Wt + EWf + - - .) + EE2(W$ + EW,2 + * - .) * - - + C.C. 

p = Po + €PI + * * - + {s'E(P,' + EPf + * - .) + E E (V,2 + EV,Z + - - .) * * * + C.C.} 

02 

02 

(3.2a, b, c, d) 

Here 'C.C.' denotes 'complex conjugate' whilst E = exp(iz/E). In the absence of a vortex 

the flow reduces to the basic state w = p = 0,u = a,v = 8. We note that the only z 

dependence in (3.2) is through E so that ~O,BO, etc. are functions of x and y only. 

The expansions (3.1), (3.2) are then substituted into (2.5) and like powers of E for 

each Fourier component are equated. The mean flow (a0,80) is determined from the 

zeroth order continuity and x momentum equations. The pressure field PO is then found 

from the normal momentum equation. The equations to determine ti0 and 80 are 

aa0 ago -+-=o, 

ax ay 
6 

Thus the boundary layer equations are now driven by the nonlinear interaction of the 

Gortler vortex with itself. The zeroth order equations for the fundamental terms in the 

core are 

Vi + GoxUiiio = 0, 

-;Po’ = w;. 

The consistency of equations (3.4b,c) requires that throughout the core 

and (3.3) then becomes 

aiio au0 

ax ay 

0, -+-= 

(3.4a, b, c, d) 

(3.6a, b) 

Thus (3.5) determines the mean flow streamwise velocity component in the core whilst 

(3.6b) then determines 1 V; 1’. It follows that in the core the boundary layer flow is now 

being forced by the vortex which from (3.6b) is itself driven by the boundary layer. This 

is, of course, the exact reverse of the roles played by these equations in the linear or weakly 

nonlinear descriptions of this problem. In fact the interaction described by (3.5) and (3.6) 

is not unlike that postulated some years ago by Malkus (1956) for turbulent flows. In the 

latter theory it was argued that the ’mean flow’ for a turbulent flow organizes itself so as 

to make any possible instability marginally stable. In the large amplitude Gortler problem 

this mean flow adjustment is achieved by centrifugal effects and is described by (3.5). We 

can integrate (3.5), (3.6) to give 

(3.7~’ b) 

I 

I 

I ‘ where a(x) and b(x) are arbitrary functions of x and a dash denotes a derivative with 

respect to 2. The function 1 V: 1’ is found by integrating (3.6) to give 

I 
7 

where B(x) is another function of z to be determined. The function 

negative so y1 and y2 which determine the edges of I satisfy (3.8) with 

then eliminated we obtain 

V, l2 cannot be 

= 0. If B(z) is 

The above equation is not of course sufficient to determine a, b, y1 and y2 so we are not yet 

able to determine the location of the lay& IIa,b. The thickness of these layers is determined by a balance between diffusion across the layers and convection in the streamwise 

direction. This balance shows that the layers are of thickness so that in IIa we write 

respectively. We can 

la a ” a and -- 

a a 

€213 at Hence in IIa we replace - and - by - - -- 

dX ay ax €2/3 at 

see from (3.8) that I Vd 12- y2 - y when y --f yz- in which case the fundamental velocity 

components and pressure in (3.2) decrease by O(dj3) where y - y2 = O(e2j3). The 

appropriate expansions in IIa therefore take the form 

The coefficient in the above expansions are functions of x and t. 

It has been anticipated above that the first harmonic functions also decrease in size 

in IIu; this decrease is forced by the form of the equations for U:, V: in the core. The 

appropriate equations are 

4U: + V2b = F1, (3.11a, b) &io 

Y 

2- -F 4vo” + GOXU0 uo - 2, 

where F1 and F2 are quadratic in Ut, V,. The forcing terms are therefore O(E~/~) in IIa 

and so U:, V:, W:, Po2 must be rescaled in that layer as indicated in (3.10). We note that 

(3.11) can always be solved in the core since ti0 satisfies (3.5). A similar analysis applies 

to the higher harmonics so we conclude that the fundamental effectively drives the mean 

flow and all the higher harmonics. The situation is quite different from the usual type 
8 

of weakly nonlinear based on the Stuart-Watson method where the mean flow remains 

essentially identical to that present before any instability occurs. 

We now return to the solution of (2.3) in the transition layer IIa. The first two terms 

in the expansion of the mean flow in this layer satisfy 

and the solutions of these equations which match with the solution in the core are 

(3.12~’ b) 

The function Z2 is forced by the fundamental terms in (3.10) so that we cannot proceed 

further without solving for the latter terms. The equations to determine (U01,Vol) and 

(U11,Vll) are found to be 

= 0, +VOl 

Goxd--%z uo 1 

(3.13~’ b, c, d) 

The equations (3.13a,b) are of course always consistent but (3.13c,d) are only consistent 

if an orthogonality condition holds. To obtain this condition we must proceed to higher 

order in the mean flow expansion. 

The first order solution for 50 is found to be 

whilst E2 satisfies 

which can be integrated once to give 

-- az2 b 2 I VOl l2 + f. (3.14) at - -I 3X2Go 2y21 + 4G4-1 - Gw 
9 

Here f(x) is another function which can only be determined at higher order; for our 

purposes here it is not required here. Having determined - we can now write down the 

appropriate solvability condition for (3.13b): 

aE2 

a€ 

where 

(3.15) 

+ bdGTdZ3Gl. 1 x’(a + 2Y2I2 

’, 3% S(x) = -[ a + 2Y2 

This equation is a particular form of the second Painleve transcendent and has been shown 

by Hastings and Mcleod (1978) to have a solution such that 

It follows that in IIa the fundamental terms decay to zero so that the finite amplitude 

Gijrtler vortex is trapped below region IIIa. We note that a similar analysis for the higher 

harmonics shows that these functions also decay exponentially to zero in IIa. However 

the mean flow is virtually unaltered by the presence of IIa, thus the first two terms in the 

expansion of the mean flow in IIa are simply obtained by expanding the mean flow in I 

in terms of €. This means that the mean flow in IIIa must to zeroth order have ii,tiy and 

8 defined by the coreflow solution evaluated with y = y2. 

An identical analysis to that above shows that the z-dependent part of the flow is 

reduced to zero exponentially in IIb. Hence in IIIb there is only a mean velocity field. 

Thus in IIIa, b we write 

= a + o(€~/~), = D + o(€~/~), = = 0, (3.16) 

so that (a, a) defined in (0, y1) and (y2,oo) satisfies 

da aa a2a 

a-+g-=- 

ax ay ay2’ 

aa ao 

-+-=o. 

ax ay 

These equations must be solved subject to 

(3.17) 

(3.18) 

(3.19) 

(3.20) 
10 

The equations (3.17) - (3.21) together with (3.9) specify a free boundary problem for y1,y2 

and the functions a(x),b(x). Clearly no analytical solution to this problem is available. In 

fact it is also necessary to specify something about the ‘upstream’ nature of the instability. 

This amounts to finding an asymptotic form for the solution of this system close to the 

value of x where the original boundary layer becomes linearly unstable. This will also be 

discussed in $5 where a large x solution of the system will also be developed. In 95 a scheme 

which we have used to solve the system numerically will be described. However there is a 

special case where a similarity solution of the system can be generated. This corresponds 

to the case when the undisturbed boundary layer is Blasius flow and x is proportional to 

Though this particular case is perhaps not of much practical importance its solution 

is instructive in that it suggests how to solve the full free-boundary partial differential 

system numerically. For that reason we now indicate how the similarity solution can be 

calculated. 

Consider then the curvature distribution x(x) defined by 

x(x) = 6. 

I 

(3.22) 

If the undisturbed flow is a Blasius boundary layer the local Gartler varies with x2 

as does the fourth power of the local spanwise wavenumber. This is consistent with the 

scaling of the right hand branch of the neutral curve discussed by Hall (1982a). Hence on 

the basis of linear theory the flow is either stable or unstable for all values of x. We define 

the similarity variable q by 

Y q=- 6’ 

and seek a solution of (3.17) by writing I 

(3.23) 

1 

fi = f‘(t7)’ zi = -{qf’- 6 f}. 

The functions a and b then take the form 

- 

(3.24) 

(3.25) 

We can then show that the free boundary problem specified by (3.9)’ (3.17) and (3.21) is 

equivalent to 

f ”’ + ff” = 0, (3.26) 

on 

subject to 

(3.27) 
11 

and 

f'(m) = 1, 

together with the matching conditions 

(3.28) 

'c 

and the scaled form of (3.9): 

Yi Here we have replaced - by qj for j = 1,2. Thus in the special case x = 6 we have 

a free boundary ordinary differential system to solve. The unknowns are ii, ?; and ql,q2, 

the boundaries of the region where a finite amplitude vortex exists. These constants and 

the function f were obtained by following the procedure shown below. 

(i) Integrate (3.26) forward from q = 0 with f(0) = f'(0) = 0 and an initial guess for 

f"(0). This integration is stopped at q = ql where f'(ql)f"(ql) = G. 

(ii) Using f(q1), f'(q1) calculate 6,6 from (3.29a,b) with j = 1. 

(iii) Now calculate q2 from (3.30). 

(iv) Finally integrate (3.26) from q = q2 to some suitably large value of q with 

6 

f (q2), f'(q2) defined by (3.29a,b) with j = 2 and using 

(v) If f'(m) # 1 we return to step (i) and alter the initial value of f" at the wall in 

The above procedure was found to converge when used with Newton's method to 

update the value of f" at the wall. We found that, as expected, a solution exists only for 

Go > 4.2 the neutral value of Go. Beyond Go = 4.2 the values of q1 and q2 respectively 

decrease and increase from their limiting value ql = q2 = 1.56 when Go = 4.2. At larger 

values of Go,q2 varies linearly with Go whilst ql goes to zero like GL3. These scalings 

can be recovered by an asymptotic investigation of (3.26) - (3.30) in the limit Go 4 00. 

some suitable manner. 
12 

Such a calculation shows that for Go >> 1, 71 = .l2GC3 and the value of 

7,9 predicted in this limit is shown in Figure 2 where we have shown the result of a full 

numerical solution of (3.26) - (3.30). 

Thus for the particular case x = the partial differential system governing the 

development of the mean flow can be reduced to an ordinary differential one. Moreover the 

results shown in Figure 2 suggest the type of behaviour for y1, y2 which we should expect 

for the general problem when x increases beyond its neutral value. At large values of x 

we shall see that a,structure essentially identical to that shown in Figure 2 for Go >> 1 

is set up. Finally we note that for other similarity boundary layers we can always choose 

a particular curvature distribution x (5) which leads to an ordinary differential system for 

the mean flow. 

4. The initial and ultimate downstream developments of the instability. 

We first describe the development of the instability close to the downstream location 

where the boundary layer (U,g,O) becomes linearly unstable. The description of the flow 

in this region is similar to that given by Hall (1982b) in the weakly nonlinear regions so 

we shall not give all the details here. Suppose then that vortex with wavenumber 

is linearly unstable for x > x* and the instability originates in a layer of thickness 

centred on y = y*. The weakly nonlinear theory of Hall (1982b) suggests that the finite 

amplitude vortex occupies a region of depth O(x - x*)1/2 for I x - x* (<< 1. We therefore 

expand y1 and y2 in the form 

72 N +Go, 

Y1 = Y* - (z - x*) 3 g + O(x - x*), 

y2 =y*+(z-x*)hg+O(x-x*), 

It is convenient for us to now define a similarly variable t by 

(4.1~2, b) 

(44 

so that 

a a €3 a la - --+--A- - j-- 

ax ax 2xac' ay X1/2d(' 

In region I we express (uo,g0) which satisfies (3.5) in the form 

is0 = B + Xu&) + - - - * (4.3~2, b) 

Here (a, a, 0) is again the boundary layer flow which exists in the absence of the instability. 

In the neighbourhood of (x*, y') ii, is, and the curvature x expand as 
13 

x = xo + xx1+ x2x2 + - * - . (4.4a, b, c) 

If we substitute (ao,Vo) from (4.3) into (3.5) and use (4.4) then, equating like powers of 

Xo, Xf, X we obtain 

GoXO~OOulo = 1, 

u:o + 2UOOU20 = 0, 

(4.5~4 b, c) 

Equation (4.5a) is satisfied if x* is the neutral location whilst (4.5b) holds because y* in 

the linear theory is chosen so that 

d da -@-) = 0, 

dY dY y = y*. 

Equation (4.5~) can be integrated once to give 

where 

and 

XlUlO ~OlUlO 

xo uoo 

A. = --Ull - - - - 

A1 = - 

UlOU20 

2100 

(4.7i.2, b). 

We have anticipated in (4.7) that UM is an odd function of t; this would otherwise have 

been determined at a later stage. 

The Gortler vortex function Vo]. in region I expands as 

and F(6) can then be found from the zeroth order approximation to (3.6b). This equation 

can be integrated once and F(6) is found to vanish at = & where 

Here C is an unknown constant of integration to be determined later. It follows from (4.1) 

and (4.8) that 

In region IIa the forcing of the mean flow ceases and I IT: I is rediiced to zero exponentially 

in the manner described in $3. It remains for us to determine the constant C appearing 

in (4.8). We can write down the mean flow in regions IIIa, b in the form 

c$- = -g, t+ = g. 
14 

It is easy to show that UM(() and VM(() are odd and even functions of ( respectively so 

it is enough for us to consider IIIu where ( 2 (+. The functions UM and VM satisfy the 

equations 

(4.9~2, b) 

and in order to match with the core solution (4.6) UM must satisfy 

Since UM must also tend to zero when 

value problem for (+. After some manipulation we find that 

+ 00 (4.9a), (4.10) constitute a nonlinear eigen- 

(4.11~2, b) 

where U(u,z) is a parabolic cylinder function. The eigenrelation for (+ is then found to 

be 

(4.12) 

and C is then given by (4.8). Moreover y1 and y2 are then known correct to order (x-x*)l/' 

The functions u(x) and b(x) are found by comparing (3.7) and (4.3) with ti, 8, x given 

by (4.4). Thus, for example, we find that u(x) is given by 

(4.13) 

for X << 1. Thus we can construct an asymptotic solution of the free boundary partial 

differential system of $3. This asymptotic solution can of course be used to start off a 

numerical solution of this system by a marching procedure. 

Next we suppose that the curvature distribution function becomes large when x + 00. 

More precisely we suppose ~(x)/xh + 00, x + 00 so that the flow is unstable on the basis of 

linear theory when x + 00. In this limit it is convenient to perform an asymptotic analysis 

directly on the solutions obtained in $3. For definiteness we assume that x = xM, M > 3 

when x + 00. 
15 

The scaling for x -+ 00 is suggested by the large Go solution discussed in $3 and can 

be inferred from (3.7). It is easy to show that a solution of the ‘upper problem’ in IIIa 

requires that y2 >> xfr and that iio = 0(1) when y = y2. Thus I must be of depth O(x); 

the core solution for y = O(x) can then be written 

(4.14) 

and if we take 

GOXM y2= -+... 

2 

50 + 1 + when y + y2-. We then find that 

+... G~M~~-~ 

80 --+ 

6 

when y -+ y2-.Thus a solution of (3.17) for (ii,8) in the upper layer IIIa takes the form 

(4.15a, b) 

the resulting upper layer problem for (Z,? does not have a solution. The depth of the 

region IIIb can now be inferred from equation (3.9). The dominant term on the right 

hand side of this equation is O(xMw1) so a balance with a comparable term on the left 

hand side can only be achieved if a +y1 - X2-3M. Thus y1 and a must be O(Z~-~~) and 

we therefore write 

2-3M + . . . Yl = YlOX 

a = aox 2-3M + . . . (4.16a, b) 

The solution for (ii,8) in IIIb can then be developed in the form 

2 

8 = -(M - 1)~~-~ + . . . , (4.17a, b) 

Here N is an unknown constant to be determined. Finally the continuity of the normal 

velocity component at y1 requires that we expand b in the form 

2 

h = h&5M + . . . . (4.1 74 
16 

The conditions (3.20), (3.21) for j = 1 then require that 

1 

- = N2Y1o, GO 

and 

(4.18a, b, c) 

It follows from (4.18a,b) that a0 = -ylo so it remains for us to determine ylo. The required 

equation follows from the zeroth order approximation to (3.9) which yields 

 GO 

YlO = - M2 

and then N and bo follow from (4.17a,c) respectively Thus in the limit x + 00 with 

x - xM, M > 1/2 the locations y1 and y2 have the asymptotic forms 

+ ... G~MX~ 

2 Y2 = (4.19a, b) 

and between these positions the mean velocity component in the x direction is given by 

the square root form (4.14). Thus for a curvature distribution x = xM the initial and far 

downstream regions where a finite amplitude vortex exists are as indicated in Figure 3 for 

A4 > 1/2. The intermediate region can only be completed by a numerical procedure of the 

type we discuss in the next section. 

5. A numerical scheme for the determination of the free boundaries y1,yZ 

We shall now outline a scheme which we have used to solve the problem specified 

by (3.17) - (3.20). For convenience we drop the ‘-’ notation for a,v and assume that a 

solution of the problem is known for x 5 5. The scheme which we have used can be used 

to advance this solution downstream to 5 + ;E for sufficiently small values of Z. In general 

the state for x 5 k must be calculated using the approach of $4 in a neighbourhood of 

where the vortices first become unstable. 

The first step in our calculation is to define a variable by 
17 

so that (3.17) is now to be solved on (0,~) in terms of 

and 00 together with ‘jump’ conditions at s = 1. Thus (3.17) is now written in the form 

with boundary conditions at 0, 

where j = 1 for < 1 and j = 2 for 5 > 1. The required boundary conditions are 

u=v=o, (=O, u+l, S+W 

and the ‘discontinuous’ conditions at 5 = 1 may be written 

(5.la, b) 

(5.3a, b, c) 

Here the f signs correspond to the limits c + 1+, 

associated with the + sign and j = 1 with the negative sign. 

completed by the jump condition (3.9). 

scheme I 

c -+ 1- and the index j = 2 is 

Finally the system is 

We first advance the solution of (5.la) for s in the range [0,1]. This is done using the 

where h is the vertical grid spacing and a tilde denotes a quantity evaluated at the position 

5 + E”. The index n refers to a quantity at the grid point 5 = nh. In order to solve the 

tridiagonal system (5.4) we must make a guess for cj and set y; = (cj - yj)/Z. When 

solving (5.4) we satisfy the required condition on u at < = 0 and (5.3b) with j = 1. The 

continuity equation (5.lb) is then discretized as 

so that 5 can be determined at 2 + Z for 0 5 s 1. However the equation for 5 is only 

of first order in s so only the boundary condition at 5 = 0 can be satisfied during this 

procedure. Thus the solution of (5.1) for 0 5 5 1 has been calculated at k + i? but, as 
I 18 

yet, (5.3a,c) with j = 1 have not been satisfied. However these conditions are now used to 

obtain an improved value of &I and a value for 6 by writing these conditions in the form 

- (6- a) flm + {ti+ 2~1}9x' C-+b= ~ 

2m 6-X 

and iterating until ii and 61 converge. Here ?, is the current guess for b at x+ E". The scheme 

used to find u, at x + Z for 0 5 c 5 1 can be applied in a similar manner to the region 

c 2 1. The u equation is solved subject to u 4 1,c 4 00 and (5.3b) with j = 2, whilst the 

21 equation is solved subject to (5.2~) with j = 2. Finally (3.9), (5.3~) are written in the 

The second of these . equations determines a new value for 52 and the first one then determines a value for b. Thus we now have values for u, 21, a, b, yl, y2 at x + E". We then repeat 

the whole procedure using the new values of Z,?,, el, 62 obtained in the first iteration until 

converged values of these quantities are obtained. 

The above numerical scheme was found to converge for sufficiently small values of Z 

the step length in the x direction. It was found that h = .l, E" = .005 gave a stable scheme 

for the cases investigated and produced values for y1,y2 and the other flow quantities 

correct to two decimal places. The first flow considered had Go = 5.,x = 6 and the 

free boundary value problem of $3 was integrated for x > .5. At x = .5 the initial values 

of y1, y2 etc. were calculated from the similarity solution of $3. In Figure 4 we have shown 

the values of y1, y2 calculated using the scheme outlined in this section. In the same Figure 

we have indicated the values of y1,yz predicted by the similarity solution of $3. We see 

that there is excellent agreement between the results from the different solution methods. 

As an example of a non 'self-similar' Gbrtler vortex we considered the case x = 

2x,G0 = 4.176. This curvature distribution has a basic state which is linearly unstable 

beyong x* = .5. The small (x - x*) asymptotic solution of $4 was used to generate initial 

values of y1, y2, a, b and the velocity profiles at x = .51. The numerical scheme described 

in this section was then used to advance the solution beyond x = .51. The results obtained 

for y1 and y2 are shown in Figure 5 for .51 e x < 1.8. In this Figure we have also shown 

the corresponding results predicted by the small x - x* and large x asymptotic solutions of 

$4. We see that, apart from an apparently constant horizontal translation, the numerical 

scheme predicts values of y1 and y2 which rapidly approach their large x amplitude values. 

This translation is not unexpected since the large x solution of $4, to the order given, has 

an arbitrary origin for x. 
19 

In Figure 6 we have shown the streamwise velocity component produced by our scheme 

for the numerical solution of the free boundary problem in $3. The Blasius flow appropriate 

to this position and the large x asymptotic solution for u are also shown. It can be seen 

that at x = 1.8 the asymptotic and numerical solutions are virtually identical. Thus at 

larger values of x we can approximate u by the asymptotic solution of $4. In Figure 6d 

we have shown the result of making this assumption to calculate u at x = 7.5'15. We see 

that by this step the boundary layer has been substantially thickened by the effect of the 

vortices. This is because the asymptotic solution has y2 - x - x whilst the undisturbed 

boundary layer grows like 24. 

In Figures 7 and 8 we have shown the eigenfunctions V: and Ut at the downstream 

locations x = .6,1.0,1.4,1.8. The corresponding large x asymptotic solutions for V: and 

Ut can be derived using the analysis of $4. For x = 1.8 such a calculation produces 

results virtually identical to the ones shown. In Figure 8 we see that when x increases the 

streamwise velocity component of theG6rtler vortex develops a shear layer near the lower 

boundary. The development of the shear layer is caused by the fact that the transition 

layer IIb approaches the wall when x --f 00. Thus Vi - v' increases rapidly from 

zero to an 0(1) value in the neighbourhood of yl. 

The implication of the above calculations is that we can reasonably expect that the 

large x solution of$4 will give accurate predictions for the vortex induced flow quantities at 

relatively small values of x. Thus in practical situations we might expect to obtain sensible 

results by using that approach rather than the numerical scheme for the free boundary 

problem. Moreover such an approach produces velocity profiles whose lower stability to 

Tollmien-Schlichting waves can be readily investigated. 

6. Conclusions 

4- 

It is perhaps useful at this stage to remind the reader that the small wavelength approximation wehave used does not make our calculation physically unrealistic. This is 

because it is known experimentally that when Gortler vortices develop their wavelength 

is conserved as they move downstream. Thus for a growing boundary layer the effective 

wavenumber increases in the downstream direction so that a small wavelength approximation eventually becomes justifiable. We have no reason to suppose that in the non 

linear case the downstream position where the small wavelength results approach the 0(1) 

wavelength results will differ significantly from that for the linear case. 

The discussion in $3-5 has been concerned with flows for which x increases at least 

as quickly as x$ when x increases. Otherwise the basic state will become linearly stable 

at a finite value of x and the structure we have found will terminate at some value of x. 

The termination of the finite interval of vortex activity is simply the 'mirror-image' of the 

small (x - m) solution of $4. This result can be confirmed from the weakly nonlinear 

theory of Hall (1982b) so that at some value 2, say x+, y1 and y2 will have the asymptotic 

form y+ f y++(x+ - 5) k where y+, y++ are constants. In Figure 9 we have shown the 

development of y1,yZ for Go = 4.176 with x = 22, x 2 1. We 

see that the region of vortex activity which begins at linear neutral position x = .5 stops 

x 5 1, x = 4x2 - 3x3, 
20 

at x = 1.5. In the absence of a finite amplitude vortex beyond z = .5 this flow is linearly 

stable for x > 1.854. Thus the presence of the vortices in the range .5 < x < 1.5 produces 

a boundary layer which is stable in a regime where it would have been unstable if the 

vortices had not developed. 

However the effect of the vortices on the boundary layer does not end when the layer 

of vortex activity terminates. This is because the initial velocity distribution at x = x+ 

for the subsequent boundary layer will in general be quite different from that appropriate 

to the undisturbed flow. Furthermore there is no reason to suppose that y+ should equal 

y* the location of the vortices according to linear theory applied to the undisturbed state. 

Thus the decay of the vortices at x = zf does not in any sense allow the boundary layer 

to return to its undisturbed state. It follows that Gortler vortices might have a significant 

effect on separation subsequent to a region of concave curvature. Indeed it is known from 

the work of Hall and Bennett (1986) that triple-deck flows can support Taylor-Gortler 

vortices so the properties of these flows might also be significantly altered in the presence 

of vortices. 

In some flows it is possible that there will be several intervals in z where vortices can 

develop. We might expect that the steady boundary layer over a wavy wall might support 

vortices at regular intervals along the wall. In the intervals where vortices do not develop 

the basic state will in general be altered from its undisturbed state by the vortex activity 

in the previous undisturbed interval. 

The most surprizing feature of our calculation is that the fully nonlinear state driven 

by large amplitude Gortler vortices can be described in a relatively simple manner. The 

major effect of the vortices is to gradually expand into the boundary layer to give the 

mean flow there a simple square root profile. If the location increases faster than xi this 

layer thickens until it occupies the whole of the boundary layer apart from a thin layer at 

the wall. In addition the layer of vortex activity expands into the free stream and thus 

thickens the undisturbed boundary layer. 

Acknowledgement 

partially carried out at Exeter with SERC support. 

The work of PH was partially carried out at ICASE. The work of WDL was 
I 

21 

REFERElVCES 

Bennett .J. and Hall P.. 19x7 : 

suhmittcd to J. Fluid 3ltch. 

Flor~-ali, J. and Saric W., 1987 : 

A.1..4.-4. pnpcr 79-1497. 

GGrtler H, 1940 : 

Tech. Mcmor. Nnf. Ado. ('omin. -4ci.o. IVcishinytoiz No. 1375. 

Hall P., 1982a : 

J. Fluid hicch. Chl. 124 p 475. 

Hall P., 1982b : 

J.I.il1.A. 101. 2.9 p 179. 

Hall P., 1983 : 

J. Fluid illcch. 1.90 11 il. 

Hall P. arid Bennett J. 19% : 

J. Fliritl .\Itch. l.01 171 p 441 

Hall P. and Sniitli F.T., 19x7 : 

siihriritttd to P1.o~. Roy. ,SOC (A4) 

Jlainiiierliii (i., 195.5 : 

J. Rct. .\lcch. -4iia1. 1-01 { p 279. 

I1ani1iierlin G., 19.56; : 

2..4.M.P. I]' 156. 

Hastings S. and Mcleotl J.B.. 1978 : 

hlR (' i*c1>od 1861 

Malkus M'.. 19.56 : 

J. Fluid illcch. C'ol 1 1) 21 

Siiiitli .L\.hI.O.. 19.55 : 

(2. .lpp/. -\l(/i//. 1'01 1 Y 1' 2J3. 
22 

X 

G 

0 

W 

n 

W 

X 

A- 

\\ \ c3 

\\ 

n 

X 

AW 

cd e M n 

U 

A1 

(11 

C 

E 

IC 

a, 

L 

c, 

m 

c 

5 

U 

(11 

L 

c, 

-0 

c 

0 

>, 

a, 

n 

m 

C 

M 

a, 

L 

c, 

c 

a, 

L 
23 

\ 

\ 

\ 

\ 

e - 

F+ I I 1 

PJ Go d 

* 
24 

Y 

Y* 

I 

I I 

I I Small 

I (x-x*) I 

I 

Linearly I , weakly I 

stable I nonlinear I I solution 

I I 

I 

I 

I I I 

Numerical 

solution Large x Solution 

I I I +x 

X* 

Figure 3. The different regions of vortex activity as the 

vortices develop in the downstream direction for 

-M x x,M>4 
Yl? Yl 

0 Numerical solution 

Similarity solution 

c 1 I I I I I ))i 

1 2 3 4 5 6 

Figure 4. Comparison between the similarity solution of 53 with the 

nurncrical scheme of 54 applied to the case x = J~x, go = .5 
26 

c 

c 

0 

a 

.* 

- 

x 

I 

I 

I 

\ 

\ 

\ 

\ 

X 

CD 

r. 

r 

dII 

0 

bo 

X 

N 

II 

X 

X 

U 

C 

rg 

L 

a, 

k 0 

a, 

> 

a, 

711 

a, 

L 

!- 
2: 

Y 

6 

5 

4 

3 

2 

1 

Blasius 

flow 

Jumerical solution 

Large x asymptotic solution 

I I I I 

Figure 6a. The streamwise velocity component 

corresponding to x = 2x, go = 4.176, x = 1. 
28 

Y 

-2 *4 

I 

flow 

U 

Figure 6b. The streamwise velocity component 

correspnr,ding to x = 2x, go F 4.176, x = 1.4 
29 

Y 

6. 

5. 

4. 

3. 

2. 

1. 

Large x asymptotic 

Blasius 

flow 

2 solution 

I I I I I 

w 

$4 -8 1. U 

Figure 6c. The streamwise velocity component 

corresponding to x = 2x, g, = 4.176, x = 1.8 
30 

Y 

60 

50 

40 

30 

20 

10 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

Asymptotic solution I 

/I 

x = 15. I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

/ I 

/ I 

/ I 

/ / 

I / 

/ / 

/ / 

/ / 

/ / 

I 

/ 

I 

/ 

/ / / 

\Asymptotic solution 

x = 7.5 

slasius flow x = 7-5 

Blasius flow x = 15. 

I I I I I *U 

Figure 6d. The streanwise velocity component corresponding to 

x = ZX, go = 4.176, x = 7.5, 15. 
31 

-0 

> 

C 

0 

h 

a, 

k 

7 
32 

1.8 

Figure 

I i 

8. The eigenfunctions U at the downstream ocation 

x = .6, I., 1.4, 1.8 $or the case x = Zx, go = 4.176. 
. 

x 
Standard Bibliographic Page 

L. Report No. NASA CR-178268 

ICASE Report No. 87-16 

2. Government Accession No. 

7. Author(s) 

Philip Hall and William D. Lakin 

I 

I 

I 

3. erforlping Org izatio Name and ddr s. Lst+tute Tor tomputer IppYications in Science 

Mail Stop 132C, NASA Langley Research Center 

Hampton, VA 23665-5225 

L2. Sponsoring Agency Name and Address 

National Aeronautics and Space Administration 

Washington, D.C. 20546 

and Engineering 

19. Security Classif.(of this report) 

Unclassified 

15. Supplementary Notes 

Langley Technical Monitor: 

J. C. South 

~ ~~~ 

20. Security Classif.(of this page) 21. No. of Pages 22. Price 

Unclassified 35 A0 3 

3. Recipient’s Catalog No. 

5. Report Date 

March 1987 

6. Performing Organization Code 

8. Performing Organization Report No. 

87-16 

10. Work Unit No. 

11. Contract or Grant No. 

NAS1-18107 

13. Type of Report and Period Covered 

Contractor Report 

14. Sponsoring Agency Code 

505-90-21-01 

Submitted to 

Proc. Roy. SOC. Ser. (A) 

16. Abstract 

The fully nonlinear development of small wavelength Giirtler vortices in a growing boundary layer is 

investigated using a combination of asymptotic and numerical methods. The starting point for the analysis 

is the weakly nonlinear theory of Hall (1982b) who discussed the initial development of small amplitude 

vortices in a neighbourhood of the location where they first become linearly unstable. That development is 

unusual in the context of nonlinear stability theory in that it is not described by the Stuart-Watson approach. 

In fact the development is governed by a pair of coupled nonlinear partial differential evolution equations for 

the vortex %ow and the mean flow correction. Here the further development of this interaction is considered 

for vortices so large that the mean flow correction driven by them is as large as the basic state. Surprisingly 

it is found that such a nonlinear interaction can still be described by asymptotic means. It is shown that the 

vortices spread out across the boundary layer and effectively drive the boundary layer. In fact the system 

obtained by writing down the equations for the fundamental component of the vortex generate a differential 

equation for the basic state. Thus the mean flow adjusts so as to make these large amplitude vortices locally 

neutral. Moreover in the region where the vortices exist the mean flow has a ‘square-root’ profile and the 

vortex velocity field can be written down in closed form. The upper and lower boundaries of the region 

of vortex activity are determined by a free-boundary problem involving the boundary layer equations. In 

general it is found that this region untimately includes almost all of the original boundary layer and much 

of the free-stream. In this situation the mean %ow has essentially no relationship to the flow which exists in 

the absence of the vortices. 

17. Key Words (Suggested by Authors(s)) 

Gortler vortices 

18. Distribution Statement 

34 - Fluid Mechanics and Heat 

Transfer 

Unclassified - unlimited 

NASA-Langley, 1987