I 

1 JOINT INSTITUYE FOR AERONAUTICS AND ACOUSTICS / &/O 3 3 7 

I 

1. National Aeronautics and 

I Space Administration I Ames Research Center 

JIAA TR- 84 Stanford University 

~ CONTRACTION DESIGN FOR 

SMALL LOW-SPEED WIND TUNNELS 

BY 

j 

James H. Bell and Rabindra D. Mehta 

I 

tt?iSA-CR-l82747) COITBICTIOl DESIGN FOR Ii88-220 45 

SfilLLL LOP-SPEED IJIID TUloloBLS [Stanford I Univ.) 37 p CSCL 148 

Unclas 

I G3/09 0140237 

Stanford University 

Department of Aeronautics and Astronautics 

Stanford, CA 94305 

APRIL 1988 
ABSTRACT 

. 

An iterative design procedure has been developed for two- or three-dimensional contractions installed on small, low-speed wind tunnels. The procedure consists of first computing the potential flow field and hence the pressure distributions along the walls of a 

contraction of given size and shape using a three-dimensional numerical panel method. 

The pressure or velocity distributions are then fed into two-dimensional boundary layer 

codes to predict the behavior of the boundary layers along the walls. For small, low-speed 

contractions it is shown that the assumption of a laminar boundary layer originating from 

stagnation conditions at the contraction entry and remaining laminar throughout passage 

through the “successful” designs is justified. This hypothesis was confirmed by comparing 

the predicted boundary layer data at the contraction exit with measured data in existing 

wind tunnels. The measured boundary layer momentum thicknesses at the exit of four 

existing contractions, two of which were 3-D, were found to lie within 10% of the predicted 

values, with the predicted values generally lower. From the contraction wall shapes investigated, the one based on a fifth-order polynomial was selected for installation on a newly 

designed mixing layer wind tunnel. 

1 
NOMENCLATURE 

C: 

C,: 

He : 

H; : 

L: 

P: 

Re: 

R: 

u,v,w: 

U, : 

'I' 21 ,v ,w: 

u,v,w: 

x, Y, z : 

.Y ' : 

v: 

P: 

L: 

__ 

,\ 0 rnaz: 

Contract ion rat io 

Pressure coefficient on contraction wall 

Contraction height at exit 

Contraction height at inlet 

Contraction length 

Static pressure on contraction wall 

Reynolds number, UL/u 

Contraction radius of curvature 

Mean velocity in the X,Y,Z directions, respectively 

Free-stream velocity in the wind tunnel test section 

Fluctuating velocity components in the X,Y,Z 

direct ions, respectively 

Instantaneous velocity in the X,Y,Z directions, 

respectively, e.g. u = U + u' 

Cartesian coordinates for sirearriwibe, ficizz!, 

Non-dimension a1 s treamw ise distance, X / L 

and spanwise directions, respectively. 

Kinematic viscosity 

Density 

Contraction match point 

(overbar) Time-averaged quantity .'---:---- at statinn lVlc%Allllulii -- _____ - 

2 
1. INTRODUCTION 

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Even with today's computers, a wind tunnel is still an essential engineering tool for 

model tests, basic experimental research and computer code validition. Since the 193Os, 

when the strong effect of free-stream turbulence on shear layer behavior became apparent, 

emphasis has been laid on wind tunnels with good flow uniformity and low levels of turbulence and unsteadiness. In the past, it has been difficult to devise firm rules for wind 

tunnel design mainly due to the lack of understanding of flow through the various tunnel 

components. The first attempt at providing some guidelines for the complete design of 

low-speed wind tunnels was that due to Bradshaw and Pankhurst (1964). However, recent experimental studies of flow through individual components of a wind tunnel (Mehta, 

1977, 1978 and Mehta and Bradshaw, 1979) have led to increased understanding and design philosophy for most of the components with the notable exception of contractions. 

Direct design methods for wind tunnel contractions are still not readily'available although 

the subject has received considerable attention over the years - about 40 papers dealing 

directly with contraction design are cited and discussed below in Section 2. 

Contraction nozzles form an integral part of almost all wind tunnels designed for fluid 

flow research. 'I'hey are normally insiaiieu upC1calll 

main purposes. Firstly, a contraction increases the mean velocity of the flow and this 

allows the honeycomb and screens to be placed in a lower speed region, thus reducing 

the pressure losses and hence the tunnei power faciGi. Secorrd!y, since the tn?d pressure 

remains constant through - the contraction, both mean and fluctuating velocity 'variations 

are reduced to a smaller fraction of the average velocity at a given cross section: this 

also rr;e;ns that, ir! pri~ciple, fewer screens would be required in the settling chamber, 

thus reducing the pressure losses even further. The most important single parameter in 

determining these eEecis is the contiactioc iatic, F,. The fac!nrs of reduction of mean 

velocity variation and turbulence intensity for axisymmetric contractions were derived 

theoretically by Batchelor (1953) using the rapid distortion theory and are given as: 

" ' ---A---- -I +I.- fie+ rartirrn 2nd serve t,wo .I . uALL tbY, V_..Y.I._ ____ 

(i) U-component mean velocity: Ijc 

(ii) V or W-component mean velocity: v'c 

(iii) u-component r.m.s. intensity, u': $ ,3(ln4cz - I);'/* 

(3c)"' (iv) v or w-component r.m.s. intensity, u' or uJ: 7 

- 

-4 contraction is less efficient in suppressing longitudinal turbulence than mean velocity 

variation. The reduction of mean velocity variation can be easily demonstrated by applying 

Bernoulli's equation to a non-uniform flow through the contraction (see Bradshaw, 1970, 

p.38). In absolute level, the lateral turbulence intensities (ti' and w') are enhanced while 

the streamwise intensity (u') is reduced. This effect is more easily understood by noting 

that the effect of the contraction is to stretch the vortex filaments with their axes in the 

streamwise direction and to compress those with their axes perpendicular to the streamwise 

direction. The reduction of fractional variation of the lateral component is much less 

than that of the streamwise component, so that it should be the desired reduction in the 

3 
former (together with the desired mean flow uniformity) which decides the contraction 

ratio. Typically, the lateral fluctuation is two or three times higher than the streamwise 

fluctuation downstream of a large contraction, although further downstream, the apuren 

turbulence soon achieves isotropy. However, the measured level of ut in the test section is 

normally higher (often by a factor of two or more) than that of v' or w' because of the 

contribution of "unsteadiness" to the streamwise fluctuation (Wood and Westphal, 1987). 

The unsteadiness, which can originate from the fan or an intermittent boundary layer 

separation, is usually in the form of a low frequency oscillation which mainly contributes 

to the measured streamwise fluctuation. It is therefore generally considered to be sufficient 

to ensure a relatively low level of u' and assume that the lateral levels will be lower. This is, 

however, by no means a necessary condition and so all three components of the fluctuation 

must be measured individually when calibrating a wind tunnel. 

In terms of the contraction ratio, constraints of space and cost also have to be taken 

into consideration. Contraction ratios of between 6 and 10 are found to be adequate for 

most small. low-speed wind tunnels - defined here as tunnels with a test section crosssectional area of less than about 0.5 m2 and free-stream velocities of less than about 40 

mls. 

Apart from the contraction ratio, another parameter that has to be selected (I priori 

is the cross-sectional shape. The corner flow in a contraction is generally more liable to 

separate due $5 the very !QW ve!~cit.ies encountered in this region. Furthermore, crossflows 

and secondary flows also tend to develop in the corners. In order to avoid these undesirable 

effects. the ideal cross-sectionai contraction shape is, therefore. circular. In iespoiise, 

many of the earlier designs were either circular or octagonal, the latter being an attempt 

to compromise between rectangular and circular. However, more recent investigations 

(_h?eh!a, 1978) have shown that even for square or rectangular cross-sections, in the absence 

of separation, the corner flow features remains localized - they do not migrate to affect 

the test wall flow over most of the span significantly - and so the cross-section shape may 

be chosen to match the other tunnel components. in particular the test section. The main 

parameters left to select then are the contraction wall shape and length. 

The design of a contraction of given area ratio and cross-section centers on the production of a uniform and steady stream at its outlet and requires the ayoidance of flow sep 

aration within it. Another desirable flow quality is minimum boundary layer thickness (in 

a laminar state) at the contraction exit. This suggests that the contraction length should 

be minimized in order to minimize the boundary layer growth. Shorter contractions are 

also, of course, desirable for saving in space and cost. However. the risk of boundary layer 

separation increases as the contraction length is reduced. Boundary layer separation in 

the contraction leads to undesirable contributions to the non-uniformity and unsteadiness 

in the exit flow, in addition to a reduction of the effective contraction ratio. The danger of 

boundary layer separation results from the presence of regions of adverse pressure gradient 

on the walls near each end of the contraction which become stronger as the contraction 

length is decreased. &ear the wide (inlet) end the streamlines are necessarily convex outward. By applying an order-of-magnitude argument to the V-component Navier-Stokes 

equation, one obtains the "centrifugal force" equation (Cebeci and Bradshaw, 1977, p.41): 
where R is the radius of curvature of the surface or strictly the considered streamline. So 

the static pressure near the walls is greater than that near the centerline and, therefore, 

greater than the average pressure over the cross-section. Thus the wall pressure increases 

from a point far upstream to a point in the wide end, before decreasing as the effect just 

described is overwhelmed by the effects of decreasing cross-sectional area. The converse 

applies at the narrow end. However, in general, the boundary layer is less liable to separate at the contraction exit, due to its increased skin friction coefficient caused by passage 

through the strong favorable pressure gradient. Also, the concave curvature at the contraction inlet has a destabilizing effect on the boundary layer, in contrast to the convex 

curvature near the outlet which has a stabilizing effect (Bradshaw, 1973). 

It is normally possible to avoid separation by increasing the contraction length sufficiently, provided it has a reasonable wall shape so that the boundary layer does not grow 

excessively near the exit. Increasing the length, however, results in increased boundary 

layer growth due to skin friction, not to mention the space/cost consideration. On the other 

hand, a contraction that is too short also suffers from some disadvantages, in addition to 

the boundary layer separation problem discussed above. For example, when a contraction 

is shortened, say by AL. the upstream and downstream distances within which local velocity profile distortions (due to streamline curvature effects) are still reiativeiy high increase. 

and the effective saving is iess ihan AL. The bcczy of velocity ncnunifernity ir! the test 

section can be estimated by solving the Laplace equation for a semi-infinite straight duct 

Hith arbitrary inlet velocity profiie (Morei, 1976). The iirsi ierrn of the sdiitiori series. 

which will predominate for large distance from the inlet plane, shows that the nonuniformity decays as erp(-27rrz/He). Thus the velocity nonuniformity decays to 10% of its 

original value by a distance of 0.37He downstream of the contraction exit. 

A design satisfying all criteria will be such that separation is just avoided (implying a 

minimum acceptable length) and the exit non-uniformity is equal to the maximum tolerable 

level for a given application (typically less than 1% variation outside the boundary layers). 

Before entering the proposed design scheme. a review of some of the past work is first 

presented in Section 2. The present computational approach is described in Section 3 and 

the results of the proposed method and design procedure are presented and discussed in 

Section 3. Thc concluding remarks are given in Section 5. 

2. LITERATURE REVIEW 

Although we have attempted to make this review exhaustive, considering the number 

of papers published on the subject, it is quite possible that some references have been 

over looked. 

In the absence of separation. the flow in a contraction is adequately described by the 

Laplace equation. The solution of this linear equation is relatively easy for simple geome5 
tries (especially in the case of axisymmetric or two-dimensional flows) and hence many 

solutions have been derived over the years. In both, axisymmetric and twedimensional 

cases, the standard technique for contraction design was to define a centerline velocity 

distribution and then calculate a set of stream surfaces from it. Any stream surface (or 

a pair of surfaces in the 3-D case) could then be chosen as the wall. Most analyses of 

axisymmetric contractions are based on a series solution of either the Laplace equation 

for the velocity potential (Tsien, 1943, Szczeniowski, 1943, Thwaites, 1946 and Bloomer, 

1956), or the Stokes-Beltrami equation for the stream function (Cohen and Ritchie, 1962), 

for a given velocity distribution along the centerline. The solution produces an infinite 

set of stream-surfaces at different distances from the axis, and the most distant one with 

tolerable pressure gradient (leading to the shortest contraction) is chosen as the wall contour. Some years later, Bossel (1969) reformulated Thwaites’ solution which made the 

method faster and easier to use while Tulapurkara (1980) made an attempt at optimizing 

the number of terms in the solution. 

Early researchers used a very strict definition of a “tolerable” pressure gradient. They 

sought stream surfaces with a monotonically increasing velocity, thus eliminating the possibility of an unfavorable pressure gradient along the wall which might cause separation. 

I 1113 ~~IlulbIull L.VU.U v..!J be h!f!!ed by a contraction of infinite length, which asymptotically approached its upstream and downstream widths. Thus, the question became one of 

choosing the points at which to “cut” the contraction and then fairing into the upstream 

and dcKiistieam secticns. Of conrse, the process of cutting produced a contraction with 

adverse pressure gradients. but this was considered unavoidable. The loglcai criterion for 

such a contraction *‘length” was one reiated to the exit fiow uniformity; Tsieii (1313) and 

Smith and M’ang (19.14) used exit flow uniformity considerations to decide where their 

contraction contours should be terminated. The Tsien design method was Further genera!ized by Rirger and Rowen (1972) who simplified the way in which the design velocity 

distribution was formulated. 

Lighthill (1945) and Cheers (1945) used a variation of this popular technique by allowing for the problems related to finite length contractions from the start. They defined the 

velocity distributions to be solved for such that some tolerable effects due to the adverse 

pressure gradients were already included. This made the blending of the chosen stream 

surface with the upstream and downstream sections much smoother. Goldstein (1945) was 

apparently the first to solve the Laplace equation for a finite two-dimensional contraction. 

He found that the resulting contraction sections had very large adverse pressure gradients near the inlet and outlet. So the problem of adverse pressure gradients existing in 

contraction sections of finite length still remained. 

These difficulties also prompted more work on the two-dimensional case in the hope 

that techniques only usable in this case would make the problem more tractable, and that 

the results would be applicable to three-dimensional contractions. Two-dimensional cases 

were usually solved in the hodograph (V, V) plane. As with other analytic techniques, a 

streamwise velocity distribution is assumed and the corresponding wall shape is then calculated. The resulting contraction can either be infinite (Libby and Reiss, 1951) or finite 

(Gibbings and Dixon, 1957) depending on the exact method used to calculate the wall 

shape and on the allowances made for the adverse pressure gradients (Gibbings. 1964). 

ml--- ----.I:*:-.. ”-,,1,4 Am ,, 

6 
Jordinson's (1961) hodograph method, for large contraction ratios, deserves special mention because he solved the flow in each end of the contraction separately, assuming that, 

in the limit of high contraction ratios, the shape of the wide end becomes independent 

of that of the narrow end (and vice versa) because the flow in between is closely radial 

without significant longitudinal streamline curvature. Jordinson found that this technique 

worked very well for c > 10. Other methods for two-dimensional contractions were also 

tried; most were modifications of techniques used on axisymmetric contractions (Swamy, 

1961, 1962, Szczeniowski, 1963, and Mills, 1968). 

Attempts to derive axisymmetric contraction shapes from two-dimensional ones have 

met with limited success. Since most contractions have relatively large contraction ratios, 

any simple scaling of cross-sectional area can lead to unacceptable wall shapes. An excep 

tion is the approximate method of Whitehead et al. (1951) for calculating the flow in an 

axisymmetric contraction once the flow in the two-dimensional contraction with the same 

wall shape is known. '4lthough it is assumed that the streamline shapes are the same in 

both cases, the velocity along the streamlines in the axisymmetric case is made to satisfy 

the axisymmetric continuity equation. 

The problem of defining an effective length for the infinite contractions produced 

by most analytic methods continued to be studied (Gibbings, i3G.1, '1966 ar;?, La:: 1964, 

1966). Gibbings and Lau determined that the most conservative condition establishing the 

"length" of a contraction was how rapidly the centerline velocity approached its asymptotic 

value at the exit. Under some cicumstances, however, veiociiy uniformity at the exit is a 

ms:e stri~gezt cc?r?dition [Lau: 1&56). Another approach to the problem was developed 

by Gay et ai. (1973)' who used a turbulent boundary layer model to predict the largest 

aiiowaole ' '- ~UV~IJC - '--..--------- 6.A.u..v. m-3 ~nt _.__ 2nd thiir _____ the shortest allowable stream surface, for 

their contract ion. 

With the onset of computers, numerical methods have been increasingiy applied io 

contraction design. Smith and Pearce (1959) gave a numerical method for calculating the 

flow (either two-dimensional or axisymmetric) within a given boundary using a distribution of sources. More recently, Laine and Harjumaki (1975) computed the potential flow in 

axisymmetric contractions by an accurate method in which the Stokes-Beltrami equation 

is solved with a finite difference approach. An alternative approach used by Mikhail and 

Rainbird (19i8) is reminiscent of Jordinson's (1961) study. Mikhail and Rainbird considered the contraction shape as a matched set of inlet and exit regions. lnlet flow separation 

was assumed to be affected mainly by the inlet region shape. with exit flow nonuniformity 

influenced mostly by the exit region shape. The method of lines was used to solve for the 

flow through the contraction section, and a family of contraction shapes was investigated 

to determine the shape giving the minimum length. Downie et al. (1984) gave a method 

for calculating incompressible, inviscid flow through non-axisymmetric contractions with 

rectangular cross-sections. Laplace's equation for the velocity potential was solved numerically using a finite difference approach. Downie et al. investigated the effects of varying 

the amount and position of curvature on the two walls and compared the computed wall 

velocity distributions with measured data for some of the cases. In general, the effects due 

to the adverse pressure gradients were less severe in the measurements than in the computations and this observation was attributed to "boundary layer effects". Watmuff (1986) 

" 
used the relaxation method to solve the stream function equation within an axisymmetric 

contraction. After validating his results, by comparison with data from a contraction section on an existing wind tunnel, Watmuff went on to examine how the contraction might 

be improved through a change of wall shape. Watmuff was able to dispense with boundary 

layer analysis by showing that his new design had lesser adverse pressure gradients than 

the existing contraction, which was known to perform satisfactorily (without separation). 

One of the first investigations where the behavior of the boundary layer in the contraction was considered quantitatively was that due to Chmielewski (1974). Since then, 

most of the studies have involved the calculation of the pressure distributions using various numerical methods and then the application of a boundary layer separation criterion 

(Borger, 1976 and Mikhail, 1979). The most popular separation criterion used has been 

that due to Stratford (1959) for turbulent boundary layer separation. 

As computer codes capable of calculating the flow within a contraction become available more readily, studies in which a computational analysis is done as part of an actual 

contraction design effort have also become more common. Two of the most complete numerical studies of contraction design are those due to Morel (1975, 1976) and Batill et al. 

(1983) [see also Batill and Hoffman, 19861. Morel (1975) starts with a given wall shape 

[formed of two cubic arcs) and solves for the ve:acities =ithi= it. )Ie s!ss r---- nhr Jpcigc ---- 

charts in which the parameters are the maximum wall pressure coefficients at the inlet (as 

an indication of the danger of separation at this end) and at the exit (as an indication of 

the non-unifcxmity of exit veiocityj. ror any CIIOIC~S of these two paiameteys (zp?Imum 

values vary with Re), the charts yield the "shape parameter? and the nozzle length, for 

this particular family of shapes. Recently, Tulapurkara and Bhalla (1988) investigated the 

!?sv I2 twr\ zyisymmet.ric contractions designed using Morel's method and concluded that 

the exit flow uniformity was better than predicted. The report by Batill et al. (1983) inciuaes computational resuics lvl a ~lll--~~AA,=,,~,",~-! ...UI1..l- - __.- 

of several different contraction ratios. The computations were performed using both, a 

singularity panel method and a finite difference method, and the predicted pressure distributions were compared with some experimental results. A set of design charts of the 

maximum wall pressure coefficients (following the style of Morel) are also provided for that 

family of contractions. 

Most of the methods described above can only be used as design aids. Very few of 

them offer concrete and direct design information. Furthermore, the application of a lot of 

these methods also requires the establishment of some predefined design criteria for which 

limited guidance is generally given. Since it is often just as easy to choose a boundary 

shape as it is to choose a suitable critexion, designers have often used the rather unscientific 

method of design "by eye". The actual wall contour shape should not in general affect the 

reduction in mean flow nonuniformity and turbulence intensity levels which is determined 

by the contraction ratio: this was confirmed for a family of axisymmelric contractions by 

Hussain and Ramjee (1976). Klein et al. (1973) also showed that the exit flow quality 

in contractions designed by eye was comparable to that in contractions designed using 

potential flow methods, such as Thwaites' (1946) method. The usual guidelines followed 

in designs by eye are to make the ends of near zero slope and the wall radii of curvature 

less at the narrow end compared to the wide end. The gradual concave curvature is to 

n L. 

'. ' - ----'A- r-- rL--- *---e;-m- mitrhorl riihic family of contractions .. 

8 
try and minimize secondary flows due to the Taylor-Gortler instability (Bradshaw, 1973). 

Typical examples of such designs (2-D and 3-D) for small wind tunnel contractions are 

given in Bradshaw (1972) and Mehta (1977). 

3. PRESENT COMPUTATIONAL APPROACH 

The proposed scheme is not really aimed at providing contraction designs directly, 

but rather at evaluating the flow, in particular details of the boundary layers emerging 

from contractions whose wall shape has been selected a priori, using for example, one of 

the techniques described above Section 2. The present computational approach consists of 

using a 3-D potential flow code to calculate the pressure and velocity distributions along 

the walls of a given contraction. The boundary layer behavior through the contraction is 

then predicted with 2-D boundary layer codes, using the cornputled pressure or velocity 

distributions as input boundary conditions. 

The performance of a given contraction section is first evaluated using a 3-D potential 

flow code (widely known as VSAERO) which is based on the siiigti’larit;. pane! methnd 

(Maskew, 1982). The version in use at NASA-Ames runs on a Cray XMP-48; another 

version is also available for use on a %licroVA,Y. Originally designed to evaluate the flow 

ar~und ccmp!ex wi~g configiirations. VSAERO has been extensively modified to cope with 

a wide variety of problems. including internal flows (Ross, et al.. 1986). 

VSAERO uses a singularity panei method LO soive the Lapiace equation. The equatler? 

is solved for both the internal and external velocity potentials, although only the internal 

flow is of interest in the present study. The user specifies the geometry of the configuration, 

which is divided into a large number of quadrilateral panels. Source and doublet distributions cover the surface of the configuration - the strengths of the distributions are constant 

on any particular panel. The singularity distributions must be specified so as to produce 

zero normal velocity on the inner surface of the configuration. In principle. an infinite 

number of combinations of source and doublet distributions could fulfill this condition and 

so another condition must also be specified to establish the distributions. To determine 

the appropriate combination, VSAERO requires that the external velocity potential be 

equal to the free-stream velocitj potential. This particular condition is chosen so as to 

make the difference between the internal and external velocities small, thus reducing the 

perturbation which the singularity distribution must provide. Once the distribution of 

singiilarities is determined. the flow velocity at points both on and off the surface can be 

calculated. The flow nonuniformity at the exit of the contraction can be found by calculating the velocity at a large number of points dcross the plaiie of the contraction exit. 

Tlie belocity OH the surface of the contraction caii also Le calculdted, and this is used as 

the external velocity input for the boundary layer calculations. 

While \.’SAERO is capable of dealing with a great many flow problems. it was orig9 
inally designed to handle external flows. Two major difficulties arise when VSAERO is 

adapted to compute internal flows. The first is a generic difficulty afflicting all surface 

singularity panel codes. The boundary condition (zero normal velocity) is actually only 

satisfied on the center of the panel. The average normal velocity on the panel may have 

some small, but nonzero, value. This means that the flow field calculated by a singularity 

panel code will not satisfy the continuity equation exactly. A more detailed discussion of 

this effect is given in Batill et. al. (1983). VSAERO tries to deal with the problem by 

slightly changing the inlet velocity to satisfy continuity. Thus, VSAERO solves for the 

flow in a slightly porous contraction section. This effect can lead to considerable error in 

the calculated flow field. Its presence is signaled by a change in the inlet velocity from 

that predicted by continuity. The inlet velocity change was never larger than 1% in any 

of the configurations computed in this study. In general, this problem can be resolved by 

using more panels in the configuration. 

The second problem also arises from VSAERO's heritage as an external flow code. 

VSAERO always sets the velocity potential in the flow region not of interest to the freestream velocity potential. In external flows, the surface velocity does not usually differ 

greatly from the free-stream velocity, and so the singularities only produce a small perturbation. In the flow through a contraction however, the veiociiy i-aiies ?y 2 fxtz eqni! 

to the contraction ratio, which is normally of order 10. The external velocity potential 

is set equal to the velocity potential at the inlet. Thus, near the exit of the contraction, 

the external veiocity is still equal to the inlet veiocily. while the hieid velocity ha be^ 

multiplied by the contraction ratio. Most of the contractions examined in this study had 

a contraction ratio of about 8:l. This means that relatively strong singularities must be 

usea. WnicIi IiidkeS the so!utic:: !PCS acc~rate than that for an external flow around an 

object of similar complexity. This problem can also be alleviated by using more panels, 

although this; of course. requires more processiiig the. 

The exact distribution of panels with which to describe a configuartion to VSAERO 

must be chosen with some care. Areas with a large variation in surface velocity should have 

smaller panels than those with little variation. The differencing method VS.4ERO uses 

to calculate velocities is adversely affected by abrupt changes in size between neighboring 

panels and by panels with high length-to-width ratios. In this respect, previous studies 

(Ross et a]., 1986 and Batill et. al., 1983) have shown that the accuracy is improved by 

including constant area ducts upstream and downstream of the contraction so that the 

transition in panel sizes may be more gradual. Furthermore. the straight sections also act 

to separate the contraction from the source and sink panels at the ends of the computational 

"box' which have (undesirable) high cross flows associated with them. Typically, the sink 

panels were placed 1.5 contraction lengths downstream of the contraction while the source 

panels were placed 0.5 contraction lengths ahead. A typical example of the panelling 

scheme used for the present investigation is shown in Fig. 1. Note that the size of the 

panels is smaller within the contraction section where the variations in surface velocity 

would be the greatest. The increase in number of panels near the contraction exit is 

mainly to try and maintain a reasonable aspect ratio of the panels in order to minimize 

some of the problems discussed above. The tested contraction shapes were similar enough 

that one panelling scheme could be applied to all of them. The original panelling scheme 

3 1.1 

10 
was also set up to allow changes in contraction length with minor modifications to the 

panelling. The basic scheme used 1010 panels to describe a contraction, and took about 

33 seconds of CPU time to run on the NASA-Ames Cray XMP-48. All the schemes 

employed took advantage of the two axes of symmetry provided by a rectangular cross 

section contraction as well as the uimage" produced by a wall in an inviscid flow. Only one 

quarter of each contraction was actually described by the panelling scheme. The panelling 

scheme is basically the same for a mixing layer tunnel with a splitter plate down the center 

of the contraction, except that now half of the contraction is described by the panelling 

scheme. A modified version of VSAERO with 315 panels failed to properly capture the 

exit adverse pressure gradient, while a version with 2600 panels did not appear to provide 

any significant increase in accuracy. 

VSAERO can also find the velocity at specified points in the interior of the contraction. 

By taking a 10 X 10 grid of points at the downstream end of the contraction, it was possible 

to calculate the velocity nonuniformity. The standard deviation of the velocities and the 

difference between the highest and lowest velocities were used as measures of the velocity 

nonuniformity. 

The calculated pressure distributions on the curved wall and centerline of a typical 

contraction are given in Fig. 2. 3; thc v;hc!e, the pressxe decreases through the strong 

favorable pressure gradient in the contraction, as would be expected. However, note the 

regions of adverse pressure gradient on the curved wall near the contraction inlet and 

outlet. These regions are riloie appiiiefit i: the wa!! ve!ncity distributions shown in Fig. 3, 

the regions of adverse pressure gradient being indicated by a decrease in wall velocity. 

The surface velocities generated by VSAERO were used as input boundary cunditizliis 

for two different boundary layer codes - one using a simple integral method to solve the 

momentum integral equation and the other employing a finite difference scheme to solve 

the boundary layer equations. 

In all cases, it was assumed that the boundary layer originates from stagnation conditions at the start of the contraction. This is a new assumption and requires some 

justification. Now in most small wind tunnels, the flow entering the contraction comes 

through a honeycomb and a series of screens (usually at least three). The effect of a screen 

on a turbulent boundary layer is to significantly reduce its thickness and turbulence stress 

levels and scales as shown by llehta (1985). The results from that investigation showed 

that a turbulent boundary layer at moderate Reynolds numbers (Re,, - 1600) was effectively relaminarized immediately downstream of the screen. Yote that the typical Reo 

encountered in small. low-speed settling chambers is likely to be lower by at least an order 

of magnitude. Furthermore, the fine scale, low-level turbulence generated in the screen 

wire wakes dissipates very rapidly (Mehta, 1985) and it is therefore not likely to trigger 

transition in the emerging boundary layer. However. it is still possible for the laminar 

boundary layer to undergo transition within the contraction. through either the effects of 

the Taylor-Gortler instabilities in the regions of concave curvature or a separation bubble, in which case transition would occur in the separated shear layer. In either case, 

11 
the strong favorable pressure gradient, encountered in contractions with reasonable area 

ratios (c - 6-10), would invariably relaminarize the boundary layer soon after. Therefore, 

the assumption of a laminar boundary layer originating from stagnation conditions at the 

contraction entrance and remaining laminar throughout was considered to be an adequate 

approximation for all cases. 

The boundary layer was calculated along the centerline (in the spanwise reference) of 

the curved wall on the contraction. For the case of a mixing layer tunnel contraction, the 

boundary layer on the splitter plate was also calculated. 

The VSAERO output was not entirely regular - glitches occured in the surface velocity 

output as a result of discontinuities across the panel boundaries. Although the irregularities 

were small, the effect of these glitches was magnified since both boundary layer codes used 

the velocity gradient as input. In an attempt to reduce these effects, the velocity gradient 

data were smoothed using an error detection routine. A typical example of the calculated 

velocity gradient and results of the smoothing procedure are shown in Fig. 4. The routine 

compares each point with a cubic spline approximation based on the neighboring points, 

without including data from the point in question. Points which differ significantly from 

the cubic spline are moved toward the approximated line. The process is iterated until 

some defined smoothness is achieved. 

The simpler boundary layer code (Thwaites method) consisted of an integral method 

to calculate the momentum thickness. The program itself is short and simple to use (see 

Cebeci and Bradshaw. 1977, p. After the momentum 

thickness is calculated the program uses semi-empirical relations for a laminar boundary 

layer in zero pressure gradient to evaluate the other boundary layer parameters. 

ror comparisoii with the Thrvaites’ rr?et,hnd, ;r relatively complex boundary layer code 

was also used. PDl4INT ;acronym for Partial Differential Method Interactive - described 

in more detail by Murphy and King (l~azll appiies ihe gciieialkd Ca!e:kir! rr?etf?nd tn a 

system of equations consisting of the momentum and continuity equations. Given the transition location, PDMINT is capable of calculating both laminar and turbulent boundary 

lavers. as well as separation bubbles. The code uses the Cebeci and Smith turbulence model 

without a correction for pressure gradient. Originally developed for transonic boundary 

layers. PDMINT turned out to be somewhat overly complex for the needs of the present 

study. 

The momentum thickness and skin friction distributions calculated using both the 

boundary layer programs in a typical Usuccessfu19 contraction section are shown in Fig. 5. 

Initially. the boundary layer grows rapidly in the inlet region where the effects of the first 

adverse pressure gradient are felt. This is soon overwhelmed by the effects of the strong 

favorable pressure gradient. resulting in a rapid reduction of the boundary layer thickness, 

before the effects due to the adverse gradient near the outlet take over. The skin friction 

coefficient remains well positive all through the contract ion. thus indicating that separation 

is not predicted for this particular design. The results from the two techniques are seen to 

compare favorably. especially for the region near the contraction outlet. The predictions 

given by the two computational methods agree to well within 10‘,6 in this region for both 

the plotted quantities. Since Thwaites’ method is a lot simpler to obtain and use, almost 

all of the boundary layer calculations presented below were computed using this method. A 

114 for a source code listing). 

---\l 

12 
negative, or very low value for the skin friction coefficient (5 0.0005) was taken to indicate 

boundary layer separation. An example of an unsuccessful design where the boundary layer 

separated near the inlet, is given in Fig. 6. Note that both boundary layer computations 

indicate that separation would occur in this particular design; the two computations also 

predict about the same streamwise location for the separation. 

4. RESULTS AND DISCUSSION 

4.1 Validation of Computational Scheme 

The combination of an inviscid panel code coupled with the boundary layer calculations provides an estimate of the boundary layer quantities in the contraction section. In 

order to validate the proposed computational scheme, the boundary layer properties in 

four contractions installed on blower-driven (open-circuit) wind tunnels were calculated 

using this method. Details of the wind tunnel contraction sections are given in Table 1. 

The two wind tunnels with splitter plates in their contractions (A and B), were primarily designed for free-shear layer research. Ct'ind tunneis A, B aid C aie !scztcd in the 

Fluid Mechanics Laboratory (FML) at NASA Ames Research Center and wind tunnel D 

is located in the '4eronautics Department at Imperial College. The actual wail shapes for 

the four Contraction sections are plotted in a norrnaiized form in Fig. 7. Two of the wa!! 

shapes were based on a 5th order polynomial while the other two were designed "by eye7 

using the guidelines described above in Section 3. 

The computed wail pressure coeiiicienis and eakiilated dist:ib=?icns cf the hniindary 

layer properties for all four contractions are given in Figs. 8-11. Boundary layer separation 

is not indicated for any of the contraction designs. The measured and caicuiaied v&ies sf 

the boundary layer momentum thickness at the exit of all four contractions are presented 

and compared in Table 2. The comparisons are made along the wind tunnel centerline at a 

short distance (typically less than 15 cm) downstream of the contraction exit. The values 

predicted by Thwaites' method for all four cases are within about 10% of the measured 

values: the typical repeatibility for the measurements in wind tunnel B, for example, was 

about 1';. The predicted values are generally lower than the measured ones because 

the 2-D boundary layer computation obviously cannot account for the weak secondary 

flow that develops along the contraction walls (Xlokhtari and Bradshaw, 1983). This 

secondary flow is a result of the lateral convergence of boundary layer fluid towards the 

local centerline forced by the converging sidewalls. This effect would therefore not occur 

in the boundary layer on the curved walls (in the streamwise sense) in a 2-D contraction 

where the sidewalls are straight. The effect of this secondary flow which is in the form 

of a pair of streamwise vortices with the "comn~ori flow" away from the wall is to thicken 

the boundary layer along the tunnel centerline (ilehta and Bradshaw, 1988). 3ote that 

the maximum difference between predictions arid experiments occurs for wind tunnel A 

which also has the maximum three-dimensionality in the contraction geometry. It is also 

worth noting that all the contractions discussed here have an aspect ratio of about five. 

It is conceivable that for tunnels with smaller aspect ratios and longer lengths, the 2- 

. 

13 
D boundary layer computations may not work as well since 3-D effects may then start 

to dominate. Comparisons for the other boundary layer quantities which are derived 

using empirical relations in Thwaites method are given in Table 3. On the whole, the 

agreement between the predicted and measured values is reasonable. It is worth noting 

that in Thwaites’ method, the momentum integral equation is reduced to a numerically 

integrable level (for 0) by relating the momentum thickness to the shape factor and skin 

friction coefficient through quasi-similarity assumptions. The relationships are nonlinear 

and so the errors encountered in predicting the momentum thickness will typically tend 

to increase for the other quantities. On the whole, however, a reasonable quantitative 

description of the laminar boundary layer is predicted using this scheme. Fig. 12 shows 

the computed results for the 40:l contraction installed on the Smoke Tunnel which is also 

located in the FML at NASA Ames. Since boundary layer separation is predicted at the 

inlet to this contraction, the computed data are not included in Tables 2 and 3. While 

we have not been able to establish clearly if separation does occur in this contraction, this 

result is perhaps not too surprising considering the relatively short length of this design 

(L/Hi = .66) for the contraction ratio (c = 40). 

4.2 Selection of Opiiniiii-il Wall Shape 

After the computational procedure was validated through comparison with experimental resuits, it was used to seiect the optimum wa!! shape fer what was to become wind 

tunnel D. Other design considerations dictated a contraction ratio of about 8 with a 2-D 

(or at most mildly 3-D) contraction. The main requirements were that a laminar boundary 

iayer ai iob Rer is &?~ITIPI! at the splitter plate edge: this implies that the contraction 

length had to be minimized. This was in addition to the usual requirements of waniirig 

to avoid boundary iayer separcriioii ~ii the wa!!s and obtaining a reasonable mean flow 

uniformity at the contraction outlet. After an extensive review of the existing literature 

on contraction design, the following polynomial shapes were selected for testing with the 

new computational scheme. 

i) Third Order Polynomial: Y(X) = H, - (H, - He)i-2(.lr”)3 + 3(X’)2] 

ii) Fifth Order Polynomial: Y(X) = H, - (H, - He)j6(X’)’ - lqX’)4 + 10(?1’)3’ 

iii) Seventh Order Polynomial: 

Y(X) = H, - (H, - H,)[-L?O(X’)~ + 70(X‘)G - 84(X’)’ 35(lY’)4j 

iv) Matched Cubics: For 0 5 x” <_ x, 

The wall shapes given by these polynomials are plotted in normalized form Fig. 13. 

The contraction area ratio and cross-section. together with the inlet and outlet dimensions 

1.1 
had already been decided by other considerations (discussed above in Section 1). The 

contraction area ratio was fixed at a value of 7.7. The optimization process, therefore, 

consisted of choosing the ideal contraction wall shape and length. 

As shown in Fig. 14, the flow uniformity at the contraction exit improves as the length 

is increased. This is not surprising since the radii of curvature decreases as the length is 

increased. The 5th order contraction shape separated near the inlet when L/H, was 0.667. 

However, boundary layer separation is also predicted if the contraction is too long (L/H, 

= 1.79). This separation occured due to the region of adverse pressure gradient near the 

contraction ezit. A contraction that is too long allows the boundary layer to thicken (or 

its thickness is not reduced as much) and hence makes it more susceptible to separation 

near the exit. If the design is too short then boundary layer separation would also tend 

to occur, but this time near the inlet. In the present design, the main objective was to 

minimize the contraction length so that a minimum Reo would be obtained at its exit. 

After running computations on a variety of designs, it was found that the minimum 

acceptable L/H, for our requirements was 0.89. This particular length to height ratio 

was therefore used as a test case for all the contraction wall shapes and the results are 

summarized in Table 4. The 7th order and matched cubics shapes were found to separate 

near the inlet. The iiow through the 3rd aider shepe was still attached. but the flow 

nonuniformity was intolerably high, and no further optimization was attempted on this 

shape. The matched cubic shape was evaluated with the match point at x = 0.3, 0.5, 0.6, 

0.7. and 0.8. Of these cases, oniy the one with x = .3 did ne? separate; but the boundary 

layer thickness at the exit was too high giving a Red of about 600. These observations 

are in contrast to the results of Batill et al. (1983) and >lorel (1976). who found that 

the possibility of separation near the inlet decreased with increasing x. However, their 

criterion for separation was very different to ours. They applied Straford’s criterion for 

turbulent boundary iayer scpaiatioc xhich wr\n!d ehviously accomodate higher adverse 

pressure gradients, although they applied it using the corner flow parameters. Also, the 

present criterion is sensitive to the location of the adverse pressure gradient, in addition to 

the magnitude. For example, if a long straight section is inserted upstream of the region 

of adverse pressure gradient, then the present scheme would account for the additional 

boundary layer growth which may make it more susceptible to separation. 

The contraction wall shape satisfying most of the requirements discussed above is 

clearly the one given by the 5th order polynomial. It was free of separation (both on 

the centerline and in the corners) and ga\e a reasonable Ret, (- 400) and flow uniformity 

(better than 1% variation) at the operating conditions (U, = 15 rnis). This wall shape 

was therefore used on the contraction for the mixing layer wind tunnel (B). 

5. CONCLUSIONS 

A scheme is proposed for the design, or more precisely for the prediction of performance, of small, low-speed contraction sections. The proposed procedure consists of 

calculating the wall pressure distributions, and hence the wall velocity distributions, using 

a 3-D potential flow method. Although a panel method was used in this investigation, 

15 
in principle any potential flow solver should be acceptable. Once the wall pressures and 

velocities have been obtained, the boundary layer behavior can be adequately calculated 

rather than relying on some separation criteria based on the pressure coefficients, as has 

been the normal practice in the past. For the family of contractions discussed in this note, 

the assumption of a laminar boundary layer originating at the contraction entrance and 

remaining laminar in passage through it seems justified. The measured boundary layer 

momentum thicknesses at the exit of four existing contractions, two of which were 3-D, 

were found to lie within 10% of the predicted values, with the predicted values generally 

lower. Although more sophisticated boundary layer codes could be used, the present results indicate that the relatively simple Thwaites integral method is probably adequate 

for most purposes. If the prediction accuracy of within 10% on 6 is acceptable, then the 

present results also suggest that an iterative process for the wall pressure computations, 

accounting for the boundary layer displacement thickness, is not necessary. From the contraction designs investigated, the wall shape based on a 5th order polynomial was found 

to perform optimally in terms of avoiding separation and giving minimum Re@ and flow 

nonuniformity. 

Until further data are obtained, all these conclusions should be confined to small (test 

section area 5 0.5 rn-j, iow-speeu ' \ue lr' A ' *W An - 1 1 -nt-rtinnc L.vIIyA ---.____ wit,h area ratios of around -\ . eight, exit aspect ratios of about five and length to inlet height ratios of about one. 

ACKNOWLEDGEMENTS 

We are grateful to Lawrence Olson for making arrangements which allowed us to use 

me ' *'- \ 3Aljnu A pnnCUI~:~~~~~ ----**+-* n--r=rn y.''Vb.-.-- 2nd ____ to Dale Ashby for the considerable help in running it. 

We would also like to thank John Murphy for giving us access to the PDMINT boundary 

layer code and the help and advice in using it. We are indebted to thank Russell Westphal 

and David Wood for providing the boundary layer data for two of the wind tunnels. 

This work was supported by the Fluid Dynamics Research Branch, N.4SA Ames Research 

Center under grant NCC-2-294. 

REFERENCES 

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Batchelor, G. Ii., "The Theory of Homogeneous Turbulence," Cambridge 1-niversity Press, 

1953, pp 68-75. 

Batill, S. M., Caylor, M.J. and Hoffman, J.J., "An Experimental and Analytic Study of 

the Flow in Subsonic Wind Tunnel Inlets,? Air Force Wright Aeronautical Laboratories 

TR.-83-3109, October 1983. 

16 
Batill, S. M. and Hoffman, J.J., “Aerodynamic Design of Three-dimensional Subsonic 

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Bloomer, N.T., “Notes on the Mathematical Design of Nozzles and Wind Tunnel Contractions,” Aeronaut. Q., Vol. 8, 1957, pp. 279-290. 

Borger, G.G., “The Optimization of Wind Tunnel Contractions for the Subsonic Range,” 

NASA TT F-16899, March 1976. 

Bossel, H.H., “Computation of Axisymmetric Contractions,” AIAA J., Vol. 7, No. 10, 

1969, pp 2017-2020. 

Bradshaw, P. and Pankhurst, R.C., “The Design of Low-Speed Wind Tunnels,” Prog. 

Aerospace Sci., Vol. 5, 1964, p. 1. 

Bradshaw, P., “Experimental Fluid Dynamics,” Pergamon Press, Oxford, 1970. 

Bradshaw, P. “Effects of Streamline Curvature on Turbulent Flow,‘ AGARDograph 169, 

1973. 

Bradshaw, P., “Two more Wind Tunnels Driven by Aerofoii-Type Centriiugai ijhei6,’’ 

Imperial College Aero. Report 72-10, 1972. 

Cebeci. T. and Bradshaw, P., ”Momentum Transfer iii Beundary hyers,” Hemisphere 

Publishing Co.. 1977. 

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and Memorandum Yo. 2137, March 1945. 

Chmielewski, G.E., “Boundary-Layer Considerations in the Zesign oT Aeidyiiamic Csntractions.” J. Aircraft. Vol. 11. No. 8, 1974. pp. 435-438. 

Cohen, M. J. and Ritchie, 3. J. B., “Low-speed Three-dimensional Contraction Design,” 

J. Roy. Aeronaut. Soc., No. 66. 1962, p. 231. 

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Tunnel Contractions.” Aeronaut. J., Vol. 88, 1984. pp. 287-295. 

I 

Gay, B.. Spettel, F., Jeandel, D. and Mathieu. J., “On the Design of the Contraction 

Section for a Wind Tunnel,” ASME J. Applied Mech., Series E. Vol. 40, No. 1. 1973. pp. 

309-3 1 0. 

Gibbings. J. C. and Dixon, 3. R., “Two-dimensional Contracting Duct Flow,” Q. J. Mech. 

Appl. .\.lath., Vol. IO. Part 1, February 1957, pp 24-41. 

Gibbings. J. C., “Flow in Contracting Ducts,“ .4lAA J., Vol. 2, No. 1. January 1964, pp. 

19 1- 192. 

Gibbings, J. C., “The Choice of a Hodograph Boundary for Contracting Ducts,” J. Roy. 

Aeronaut. SOC., Vol. 68, No. 6, June 1964, pp. 420-432. 

17 
Gibbings, J. C., “On the Effective Length of Two-dimensional Contracting Ducts,” J. Roy. 

Aeronaut. SOC., Vol. 70, No. 6, June 1966, p. 676. 

Goldstein S., “Notes on the Design of Converging Channels,” Aeronautical Research Council Reports and Memorandum No. 2643, March 1945. 

Hussain, A.K.M.F. and Ramjee, V., “Effects of the Axisymmetric Contraction Shape on 

Incompressible Turbulent Flow,” ASME J. Fluids. Engng., Series I, Vol. 98, 1976, p. 58. 

Jordinson, R., “Design of Wind Tunnel Contractions,” Aircr. Engng., Vol. 33, No. 392, 

October 1961, pp. 294-297. 

Klein, A., Ramjee, V. and Venkataramani, K.S., “An Experimental Study of the Subsonic 

Flow in Axisymmetric Contractions,” ZFW, Vol. 21, No. 9, pp. 312-320. 

Laine, S. and Harjumaki, J., “A Theoretical Study of Axisymmetric Contractions for Lowspeed Wind Tunnels,” Helsinki University of Technology, Laboratory of Aerodynamics, 

Report No. 75-A2, 1975. 

Lai~~ W. T. F.: “An Analytical method for the Design of Two-dimensional Contractions,” 

J. Roy. Aeronaut. SOC., Vol. 68, No. 1, January 1964, pp. 59-62. 

Lau, W. T. F., “On the Length of Two-dimensional Contractions,” J. Roy. Aeronaut. 

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Libby. P. and Reiss. H., ”The Design of Two-dimensional Contraction Sections,” Q. J. 

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Lighthill, $1. J.. “A Method of Two-dimensional Aerodynamic Design.“ Aeronautical Research Council Reports and Memorandum No. 2112, Aprii i945. 

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1982. 

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Prog. Aerospace Sei.. Vol. 18, No.1, 1977$ pp.54120. 

I 

Mehta, R. D., “Aspects of the Design and Performance of Blower Wind Tunnel Components,” Ph.D. Thesis, Department of Aeronautics, Imperial College, University of London, 

Sovember 1978. 

Mehta, R.D. and Bradshaw, P., “Design Rules for Small Low Speed Wind Tunnels,” Aetonaut. J.. k’ol.83. 30.827. Xovember 19T9, pp. 442-419. 

Mehta, R.D.. ”Turbulent Boundary Layer Perturbed by a Screen,’ AIAA J., Vol. 23, No. 

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Mehta, R.D. and Bradshaw? P., “Longitudinal Vortices Imbedded in Turbulent Boundary 

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18 
1988, pp. 529-546. 

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Paper 78-819, 1978. 

Mikhail, M.N., "Optimum Design of Wind Tunnel Contractions," AIAA J., Vol. 17, No. 

5, 1979, pp. 471-477. 

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Layers," Aeronaut. J., Vo1.87, 1983, pp. 233-236. 

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Sci.. k-01. 28, No. 6, June 1961. pp. 500-501. 

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8, October 1943, p. 311. 

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AlAA J. Vol. 1, No. 4, April 1963, p. 977. 

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19 
Council Reports and Memorandum No. 2278, 1946. 

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of Aeronautics, Imperial College, London University, January 1980. 

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20 
d 

c: 

v) 

9 

d 

0 

cr! 

r 9 r 

x I-- 

Q, 

0 

oq 

z 

0 

u+ 

CW 

F x- 

$3 Eo F 

Q, 

0 

2 

4 

(3 

u! 

b Q, 

0 
moo 

No0 

UN 

00 

99 

CDN do 

00 

99 

-0 

W s3u “8 

0 

0 6m 

9 

zo 
00 

mo 

2s 

Nm 

NN 

Lncq 

am 

NN 

mcq 

I gg 

mm 

99 

om mhl ab) 88 

00 

oco mb 22 do IO0 

m 

W 

J 

m 

P 

W 

J 

3 

> 

n 

W 

IP 

a 

I! 

a 

W 

0 

II 

0 

W 

.. 

3 

a 

n 

a 

2 

a 

> 

W 

W 

II 

I 

B 

I0 

m 

N 

4 

a 

ti 

P 

W 

I- 

- 

LL 0 n 
TABLE 4 

COMPARISON OF WALL SHAPE PERFORMANCE 

CONTRACTION 

WALL SHAPE 

EXIT PLANE 

SEPARATION Reo AT UNIFORMITY 

PREDICTED CONTRACTION EXIT (STANDARD 

DEVI AT ION ) 

j NO 1 3RD ORDER 

POLYNOMIAL 369 I !?.n1n2 I I I 

I 

I NO 5TH ORDER I 

POLYNOMIAL I 425 I 0.0040 I 

I I 

YES I 7TH ORDER 

POLYNOMIAL 478 I 0.0024 

I SYMM ETR i2 

MATCHED CUBICS I YES I 0.0024 I 

CONTRACTION RATIO = 7.7, LENGTH TO HEIGHT RATIO = 0.89, 

FREE-STREAM VELOCITY = 15 m/sec 

c 

-.2 

a0 

0 

G 

2 

w .4 

z .2 

00 

LL 

LL 

u 

K 

3 

v) 

v) 

W 

K 

Q .8 

1 .o 

-.5 0 .5 1 .o 1.5 

X/L 

Fig. 2. Typical caicalated wa!! presslire distributions. 

1.2 

1 .o 

.8 

.6 

.4 

.2 

0 

-.5 

CURVED WALL 

0 .5 

X/L 

1 .o 1.5 

Fig. 3. Typical calculated wall velocities. 
v) 1.5 

3 

0 

'13 

\ --- RAW DATA 

- SMOOTHED DATA 

LEI 

> OW'' 

I I I 1 

NORMALIZED DISTANCE ALONG WALL, S/L 

1 

.4 .8 1.2 1.6 2.0 

3 -.= 0 

Fig. 4. Example of smoothing used on velocity gradient data. 
E ". .008 

z 

0 

n 

a .oo6 

K 

> .004 

a 

TH WA I TES PDM INT 

g ,002 

a 

n 

30 z 

-.25 0 .25 -50 .75 1.0 1.25 1.50 

X/L 

B 

Fig. 5. Calculation of boundary layer development through a contraction. 

7 .015r 

0 

F I 

E ". .010 

0 

n 

a 

z 

6- .005 

K 

K 

a 

n 

w 

> 

-I 

2' 0 

a 

I I 1 1 

.4 .8 1.2 1.6 

z 

3 

-.005 

0 

XIL 

Fig. 6. Boundary layer calculation indicating separation. 
WIND TUNNEL 

*- I 

.6 I .- 

I 

> 

\ 

.1 

.2 

.I 

3 0 .& .4 .6 

X/L 

.8 1 .o 

Fig. 7. Wall contour shapes of contractions used for verification of the scheme. 
cP 

0 

.2 

.4 

.6 

.8 

1 .o -.5 0 .5 1 .o 1.5 

.012 

.010 I. 

c 

b 

.008 

X 

.002 GO 1 

XIL 

THWAITES METHOD 

oe 

0 Cf 

1 1 1 1 

.4 .a 1.2 .16 

XIL 

Fig. 8. Wall pressure distributions and boundary layer calculations for 

the contraction on wind tunnel A. 
n .5 1 .o 1.5 E -. d 

X/L 

.010 

m 

p .004 

z 

a 

O' .002 

0 

THWAITES METHOD 

a6 

0 Cf 

- 

- 

- 

- 

- 

U 1 

0 .4 .8 1.2 1.6 

XIL 

Fig. 9. Wall pressure distributions and boundary layer calculations for 

the contraction on wind tunnel B. 
-.2 

0 

.2 

cp .4 

.6 

.8 

PLANE OF 8 

-.5 0 .5 1 .o 1.5 

XIL 

.010 - 

- .008 - 

z 

F 

I 

5 .006 

E 

0 .004 

z 

- 

0 

e - 

a 

cr .002 - 

THWAITES METHOD 

oe 

0 Cf 

0 

0 .4 .8 1.2 1.6 

X/L 

Fig. 10. Wall pressure distributions and boundary layer calculations for 

the contraction on wind tunnel C. 
x/ L 

,014 

-- .012 

x .010 

I 

0 

r 

- 

.008 

.006 

0' .004 

0 

a 

THWAITES METHOD 

oe 

0 Cf 

.002 

0 

0 .4 .8 1.2 1.6 

X/L 

Fig. 11. Wall pressure distributions and boundary layer calculations for 

the contraction on wind tunnel D. 
-.2 

0 

.2 

cp .4 

.6 

.8 

1 .a 2Q 

-.5 

.08 

.06 

.04 

.02 

0 

-.02 

CURVED 

0 1 .o 1.5 

XIL 

THWAITES METHOD 

06 

0 Cf 

I I I I I 

0 .4 .8 1.2 1.6 

XI L 

Fig. 12. Wall pressure distributions and boundary layer calculations for 

the rnntrartinn nn tho Smnto TPlinnol 
WALL CONTOUR SHAPE 

--- 3RD ORDER POLYNOMIAL 

. . . . . . . 5TH ORDER POLYNOMIAL 

--- 7TH ORDER POLYNOMIAL 

- SYMMETRIC MATCHED CUBIC 

I 

I I I I 

-1 0 .25 .50 .75 1 .OO 

X/L 

Fig. 13. Wall contour shapes of contractions tested. 
1.6 

s 

1.2 

.8 

.4 

0 

.6 

0 ATTACHED 

BOUNDARY LAYER 

BOUNDARYLAYER 

0 SEPARATED 

1 .o 1.4 

L/H j 

1.8 

Fig. 14. Effect of contraction length on flow uniformity.