DEPARTMENT OF MECHANICAL ENGINEERING AND MECHANICSCOLLEGE OF ENGINEERING AND TECHNOLOGYOLD DOMINION UNIVERSITYNORFOLK, VIRGINIA 23529

FINITE ELEMENT METHODOLOGY FOR INTEGRATEDFLOW-THERMAL-STRUCTURAL ANALYSES

By

Earl A. Thornton, Principal Investigator

R. Ramakrishnan, Research Associate

and

G. R. Vemaganti, Graduate Research Assistant

Progress ReportFor the period ended February 29, 1988

Prepared for theNational Aeronautics and Space AdministrationLangley Research CenterHampton, Virginia 23665

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UnderResearch Grant NSG 1321Allan R. Wieting, Technical MonitorLAD-Aerothermal Loads Branch

Submitted by theOld Dominion University Research FoundationP. 0. Box 6369Norfolk, Virginia 23508

August 1988

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FINITE ELEMENT METHODOLOGY FOR INTEGRATEDFLOW-THERMAL-STRUCTURAL ANALYSES

ByEarl A. Thornton1, R. Ramakrishnan2,

and G. R. Veraaganti3

SUMMARY

The following papers entitled, "An Adaptive Finite Element Procedure

for Compressible Flows and Strong Viscous-Inviscid Interactions" and "An

Adaptive Remeshing Method for Finite Element Thermal Analysis," were pre-

sented at the June 27-29, 1988, meeting of the AJAA Thermophysics, Plasma-

dynamics and Lasers Conference, San Antonio, Texas. The papers describe

research work supported under NASA/Langley Research Grant NSG 1321, and are

submitted in fulfillment of the progress report requirement on the grant,

for the period ended February 29, 1988.

Professor, Department of Mechanical Engineering and Mechanics, Old DominionUniversity, Norfolk, Virginia 23529.

2Research Associate, Department of Mechanical Engineering and Mechanics, OldDominion University, Norfolk, Virginia 23529.

3Graduate Research Assistant, Department of Mechanical Engineering andMechanics, Old Dominion University, Norfolk, Virginia 23529.

AN ADAPTIVE FINITE ELEMENT PROCEDURE FOR COMPRESSIBLSTRONG VISCOUS-INVISCID INTERACTIONS

R. Ramakrlshnan*Earl A. Thornton"

Old Dominion UniversityNorfolk, Virginia

and

AJUrYleting***NASA Langtey Research Center

Hampton, Virginia

•FLOWS WITH

ABSTRACT

The use of an adaptive mesh refinementprocedure for analyzing high speed compressible flowswith strong inviscid-viscous interactions is described.The adaptation procedure which uses both quadrilateraland triangular elements, is implemented with the multi-step Galerkin-Runge Kutta scheme. Elements that lie inregions of strong gradients are refined based on inviscidand viscous indicators to obtain better definition of flowfeatures. The effectiveness of the finite elementprocedure is demonstrated by modelling Mach 11.7 flowover a 15 degree ramp. Numerical results are comparedwith predictions of the strong interaction theory andexperimental data. The influence of factors such asmesh spacing and numerical dissipation on theaccuracy of computed results is discussed.

NOMENCLATURE

A element area

C Chapman-Rubesin parameter eq. (33)

Cf skin friction coefficient

CH heat transfer coefficient, Stanton number

Cp pressure coefficient

C1 C4 Runge-Kutta constants

dl • d2 artifical dissipation fluxes

EI , E2 flux components

e element error

pa advective flux

pv viscous flux

Fh heat flux

•Research Associate, Department of Mechanical Engineeringand Mechanics, Member AIAA

••Professor, Department of Mechanical Engineering andMechanics, Associate Fellow AIAA

••'Head, Aerothermal Loads Branch, Structural MechanicsDivision. Member AIAA

G growth factor, eq.(35)

i unit vector defining velocity change, eq. (7)

It. 12 components of unit normal vector

M Mach number

[M] mass matrix

[N] element interpolation function

n unit vector normal to surface

n x. n y components of unit vector n

qn heat flux normal to surface

p pressure

Pr Prandtl number

R global nodal residual

r coordinate index , eq. (10)

Re Reynolds number

S solution domain boundary

T temperature

T x, Ty surface tractions

t time

t unit vector normal to i

U typical conservation variable

u, v velocity components in coordinatedirections

xi, X2 coordinate directions

Y maximum element error

p key variable for refinement

a threshold value for refinement

1

8

u

5jj

Y

M-2, H4

TW

X

Ax, Ay

Subscripts

aw

d

i

j

o

res

w

element refinement indicator eq. (3)

Courant number

threshold value for derefinement, gridstretching parameter

Kronecker delta

ratio of specific heats

dissipation constants

shearing stress

interaction parameter, eq. (33)

grid spacings in coordinate directions

adiabatic wall

dynamic conditions

summation index

index for nodes

total conditions

reservoir conditions

wall quantities

freestream conditions

Superscripts

k index for multistep scheme

n time step

INTRODUCTION

The potential role of computational fluid dynamics(CFD) in the design and analysis of high speed vehiclesis being clearly defined by the national aerospaceplane, which is envisaged to have applications at flightspeeds exceeding Mach 25. Computational techniquesthat provide good understanding of the flow features andaccurate estimates of aerothermal loads is essential inthe design of such vehicles since ground based facilitiescannot simulate the entire flight envelope. Description ofthe complex inviscid-viscous interactions with vehiclesurfaces is possible with the full Navier-Stokesequations. At the Aerothermal Loads Branch at NASALang ley Research Center, finite element methodologyis being developed for integrated fluid-thermal-structural

analysis which can accurately predict the heating rates,pressure loads and the thermal/structural response.

Compressible flows may contain discontinuitiessuch as shocks as well as regions of high gradientssuch as boundary layers and shear layers which need tobe adequately resolved. Since these flow phenomenaoccur over small distance scales which are not knownapriori, the computational mesh needs to be adapted tomodel such high gradient regions without the use of anexcessive number of elements in low gradient regions.Adaptive mesh refinement procedures can be usedeffectively to resolve details in the flow region andminimize elements elsewhere. Such procedures lead tohighly unstructured meshes. Since finite elementmethods are characterized by their ease in handlingcompletely unstructured meshes and their ability toinclude mesh refinement procedures these methods arepursued. Explicit finite element schemes have alsodemonstrated their capacity to produce good results fora variety of flow situations and configurations1'4-

The purpose of this paper is to describe theapplication of an adaptive refinement procedure topredict viscous compressible flow features. Adaptivemesh refinement procedures for compressible highspeed flows are of recent origin. Mesh refinementprocedures for triangular finite element meshes wereinitially detailed by Zienkewicz, Lohner, and Morgan^,and the application of these procedures to steady6, andtransient' compressible flow problems has beendemonstrated extensively. Adaptive procedures forfinite element meshes with quadrilateral elements havebeen developed by Oden et. al& and by Shapiro andMurman9. A mesh regeneration scheme developed byPeraire, et. al.10 has found application in generatingmeshes comprised of triangles (in 20) and tetrahedrons(in 3D) for inviscid flows.

Most of the research in adaptive finite elementmethods has been in inviscid flows. Application ofadaptive procedures to model complicated high speedviscous flows are virtually non-existent. A notableexception has been the recent effort to use an upwindfinite element formulation11 and unstructured triangulargrids to detail complex flows resulting from shockinteraction on a cylindrical leading edge.

The present study extends the adaptiverefinement procedure and finite element formulation ofreference 12 to predict inviscid and viscous flow detailsfor hypersonic flows with strong inviscid-viscousinteractions.

MESH ADAPTATION

The classical finite element mesh refinementscheme is the addition of elements in regions of highgradients. Elements that lie in these regions are dividedinto smaller ones by a subdivision process. Bothtriangular and quadrilateral elements can be enrichedby adding a central node and/or midside nodes. Figure1(a) illustrates a triangular mesh that has been locallyenriched to capture an oblique shock. For a typicalquadrilateral element such a subdivision can result in

the generation of four smaller elements with thepossibility of the presence of midside nodes not beingconnected to neighboring elements. These midsidenodes are sometimes called "hanging nodes." Figure1 (b) illustrates a mesh with hanging nodes. The solutionalgorithm must be modified to account for hangingnodes, typically by introducing constraint equations.One way to avoid these unconnected nodes is totransition from a crude quad mesh to a fine one usingtriangular elements. Figure 1(c) shows this type ofrefinement which needs no special constraint equationsand the procedure provides a "natural" way to transitionbetween meshes of different density. Automation of theadaptive procedure is accomplished with refinementindicators.

Refinement Indicators

The rationale for using refinement indicators isthat, while it is possible to predict the location andstrength of shocks, boundary layers, etc. for some simpleflow situations, the analyst in general will not have priorknowledge of the location of regions containing sharpchanges in flow variables. The decision to refine aparticular region of the mesh can be based on eithera-priori or a-posteriori error estimates. The procedureadopted in this paper is to complete an analysis on agiven mesh and then refine the mesh at a certain stagein the analysis. The refinement at this stage is based oninviscid and viscous "error" indicators computed on theinitial mesh. The new mesh is used for the subsequentanalysis and then further refinements are performed ifneeded.

The aim of the adaptive mesh refinement is theminimization of maximum errors that occur in the finiteelement domain. The mesh requirement is then

minimize (max ej) 1 < j < n 0)

where i denotes the element number, n the number ofelements in the domain and ej the error in the i-thelement. An optimal mesh then satisfies the equaldistribution condition,

e. = constant (2)

which indicates that the error measure is uniformlydistributed for all elements of the finite element mesh.The adaptive mesh refinement procedure is designed toadd elements in appropriate regions to satisfy Eq. (2).Since the solution is not known a-priori, anapproximation to the error measure is computed usingthe finite element solution. If "p" is considered the keyvariable representative of overall solution behavior, thenan error measure for inviscid refinement is given by

(3)j.k-1.2

where 0jk are scaled second derivatives with a typicalterm 612 defined as

812"VU1

Here nd is the number of nodes in element i, and thecomma implies partial differentiation. The first term in thedenominator of equation (4) scales the secondderivative of the key variable to ensure that shocks ofdifferent strengths are adapted equally. This preventsthe strongest shock from being overdefined at theexpense of capturing weaker shocks. The coefficient e inthe second term is used to smooth the indicator inregions of oscillations.

The procedure adopted in this paper is to refineall elements that satisfy the criterion

e,>oY

and derefine all elements that satisfy

(5)

(6)

where a and P are preset threshold constants and Y themaximum element error over the entire domain. The keyvariable used for the inviscid refinement is typicallypressure or density and a and p are usually 0.2 and 0.1respectively.

The refinement indicator for viscous regions isbased on a variation of the explicit dissipation schemedetailed in reference 13. It uses the notion that near thewall the velocity changes within the boundary layer aremaximum in the direction normal to the predominantvelocity component. Unit vector i defines the change inthe value of the velocity vector V and is given by,

(7)

where V2 = (u2 + v2). The expression for element erroris given by

di (8)

where unit vector t is defined normal to i and isillustrated in figure 2. As with the inviscid indicator,elements with errors above a preset tolerance value arerefined while those with errors below the threshold arederefined.

Adaptive Procedure

The starting point of the adaptive procedure is a"skeleton" mesh or base mesh which contains only a fewelements. The skeleton mesh, which consists ofquadrilateral elements, needs to be completelystructured to begin the adaptive procedure. A structuredmesh implies that each interior node in the domain issurrounded by the same number of elements. Prior tothe first analysis, the mesh refinement program does an

overall refinement where each element of the "skeleton*mesh is subdivided into four quad elements. Thisprocedure is repeated a few times until the nodal densityof the resulting mesh is deemed sufficient to obtain asolution which captures the main details of the flow field.

Refinement indicators are computed based on thesolution obtained on this 'initial" mesh and elements thatneed to be refined or derefined are identified. Allelements in the mesh that have indicators above thepreset refinement threshold value are enriched whilethose elements that have values below the thresholdderefinement value are coarsened. The refinementstrategy used is such that at each mesh change, onlyone level of refinement or derefinement is permitted. Onrefinement of a typical element, the "sub-elements" thatresult could be all quads, or a combination of quads andtriangles. The number and type of the resulting "sub-elements" depends on the refinement level of elementsthat surround this element.

Figure 3 shows the elements that result in atypical refinement and coarsening procedure. Figure3(a) shows the original mesh where elements B, C, andD are to be refined. The mesh that results is seen in Rg.3(b). If on this refined mesh, element group C, whichincludes "sub-elements" C1, C2, C3, and C4, needs tobe coarsened, the mesh that results after derefinementappears in Rg. 3(c).

Boundary Layer Refinement

In addition to the refinement based on errorindicators detailed in earlier sections, the facility togenerate layers of structured quad elements at the wallis also included within the framework of the analysisprogram. This technique is developed to ensure a highdensity of nodes at the no-slip boundaries to resolvethe thin laminar hypersonic boundary layer. Theprogram divides the elements located at the wall intoa specified number of sub-elements and elements thusgenerated can be refined and coarsened at successiverefinement levels. The nodal coordinates of these newelements are defined such that the mesh generated isstretched normal to the wall. The location of the nodesalong normal lines at each wall location is obtainedfrom the expression of Roberts14,

5.=

x= (9)AxS.j = 1,nnew

y= yw + Ay6.

where nnew is the number of new elements that resultfrom the subdivision. Ax and Ay the grid spacings in thecoordinate direction, p the grid stretching parameter,and r the coordinate index given by,

where j is the index locating the node from the wall inthe normal direction. The nodal unknowns for the newgrid points at each wall location is linearly interpolatedfrom the nodal values of the parent element. Themeshing procedure clusters more nodes near the wallas p approaches unity.

FINFTE ELEMENT FORMULATION

The explicit multistep Galerkin-Runge Kuttaformulation used in reference 12 for inviscid flow isextended to model the compressible Navier-Stokesequations. The solution algorithm is applied to the flowequations in conservation form

IVI (11)

where U is the vector of variables and Ej the fluxvectors. The flux vectors can be written as

(12)

where Fa , pv and F^ represent the advective, viscousand heat fluxes respectively. The vectors of conservationvariables and fluxes are given by

FY =

f '

^VY

(13)

where p is the density, ui the velocity components in thecoordinate directions, p the thermodynamic pressure, ethe total energy, q the heat fluxes, TJJ the viscous stresscomponents and 5,j the Kronecker delta. The shearstresses and heat fluxes are given by,

(14)

where \n and X are the coefficients of viscosity, k thethermal conductivity and T the temperature. Otherconstitutive relations employed include:

r = j /nnew (10)

e = CVT + (u2+v2)/2

p = pRT

(15)

where Cv is the specific heat at constant volume and Rthe gas constant. The coefficients of viscosity are relatedby Stokes hypothesis

x-f. (16)

and n is assumed a function of temperature andobtained by Sutherland's law. The thermal conductivityis computed from the Prandtl number Pr which isassumed constant along with constant specific heats.Equation (11) is solved subject to proper initial andboundary conditions which include: (a) specification ofconservation variables on the inflow plane, (b)imposition of no-slip conditions and specifiedtemperatures on aerodynamic surfaces, and (c) outflowsurface integrals provided by the finite elementformulation.

Galerkln-Runge Kutta Algorithm

The system of conservation laws shown in Eq.(11) can be written as,

(17)

Application of the method of weighted residuals resultsin the finite element equations,

(18)

where [N] is the element weighting and interpolationfunction and A the domain of interest. Integrating byparts on the RHS terms yields,

[M] {^-} = /{N.j} [N] dA {E,} -J{N} [N] dS [l,EJ (19)

where the fluxes Ej are interpolated the same way as theconservation variables'.

The time marching scheme is similar to the multi-step time integration scheme which appears inreference 9 and can be written as,

U<k)=Un+ck [rrJ- R (UM) + D] k=1 ,..,41 ' lMa iv J (20)ii

un+1=u(4)

where the dissipation operator D can be written as,

(21)

and D2 and 04 are second and fourth differencedissipation terms which may be "frozen" for eachtimestep.

The second difference artificial dissipation term isneeded to stabilize solutions in the presence of sharpgradients. This dissipation operator is given by,

and

d. = KU, (i not summed ) (22)

where kj control the amount of dissipation added andare given by,

I P.. I ( i not summed ) (23)

where u.2 is the second difference dissipation constantand p the pressure. The first derivatives of pressure arecalculated as element quantities and nodal secondderivatives are then obtained from the relation,

lp,xxdA !

where the comma implies differentiation.

(24)

The linear fourth difference operator is needed todamp out spurious oscillations and provide backgrounddissipation and is given by,

(25)

The fourth differences are evaluated by repeatedapplication of a 9-point Laplacian stencil. Typical valuesof the dissipation constants na and H4 are 1/10 and1/200 respectively.

Calculation of Wall Coefficients

Heat transfer, skin friction and pressurecoefficients are essential for design considerations sincethey provide a direct measure of the severity ofaerodynamic loads on flight surfaces. Experimentalmeasurements for hypersonic flow behavior yieldsurface distributions for pressure loads, heat transferrates and skin friction coefficients. Accurate computationof the wail coefficients in the finite elementmethodology is essential since their comparison withexperimental data forms the basis for code validation.

The heat transfer and shear stress coefficientsare related to the gradients of temperature and fluidvelocity at the surface. With a structured computationalgrid it is possible to calculate derivatives at the wall bysimple differencing techniques and obtain results ofdesired accuracy. For completely unstructured finite

element meshes the calculation of wall derivatives bydifferencing is rather complicated. Computation of heatfluxes and shear stresses at the wall is possible in thefinite element context by using the "consistent*calculations. These calculations, introduced inreference 13, use equation (19) as the basis forcomputing wall coefficients.

The finite element equations (eq. 19) can be written as

J.i}[N]dA{E}-J{N}[N]dS[l.Ejs] (26)

At steady state the transient terms approach zero,implying

J{N} [N] dS [ IE J =J{N,.} [N] dA {E.) (27)

The x and y momentum equations for elements on thesurface reduces to.

(28)

.. - (e + p/p)J

J{N}[N]dS{T}

where Tx and Ty are the surface tractions and RX andRy the residuals for an element. These equations areassembled for all the elements that lie on the surface.Diagonalizing the matrices defined by the surfaceintegrals in equation (28) yields an explicit procedure forobtaining nodal values for the surface tractions. Thesurface tractions are related to the stress tensor by therelation,

{ n } (29)

where TJJ are components of the stress tensor and nx

and ny are components of the unit vector n, normal tothe surface. The wall shearing stress is then given by,

(30)

Considering the energy equation for elements that lie onthe surface, at steady state the finite element equationscan be written as,

J{N}[N]dS{qn} (31)

where qn is the nodal heat flux for an element.Diagonalizing the coefficient matrix and assembling theelement equations yields nodal values of the heat fluxon the surface. The coefficients of heat transfer (C(H).skin friction (Cf), and pressure (Cp) are computed fromthe relations,

(32)

where p« , p« andthe wall pressure.

are freestream values and p* is

COMPUTATIONAL RESULTS

The ability of the adaptive finite elementprocedure to predict flow details for viscous flows isillustrated by modelling the hypersonic flow over a ramp.This problem has special interest due to its applicationin the design of control surfaces for high speed vehiclessuch as the aerospace plane. Theoretical estimates andexperimental results are available to gauge theaccuracy of the finite element calculations.

Flow Description

The problem considered is illustratedschematically in Figure 4. Inflow at Mach 11.68 interactswith the leading edge producing a shock due to theboundary layer displacement effect. The boundary layerdoes not have enough momentum to overcome theadverse pressure gradient generated by the rampresulting in flow separation and recirculation near thecorner. The separated boundary layer then reattachesdownstream of the corner with the surface pressurerising through the separated and reattachment regions.The compression fan generated in the separation zoneeventually coalesces downstream to form the inducedshock. This shock interacts with the leading edge shockto produce a resultant shock, an expansion fan and ashear layer or slip line.

The interaction between the inner viscous shearlayer and the outer inviscid flow can be categorizedbased on the nature of coupling. At the leading edge ofthe plate the induced shock and the boundary layer areessentially of the same order and the effects of theinteraction can be reasonably predicted by variousinteraction theories. At the corner the separation resultsupstream of the induced shock due to the couplingbetween the shock and the thickening of the boundarylayer due to compression. This type of interactionnecessitates the need for solution of the full Navier-Stokes equations.

Strong Interaction Theory

Near the sharp leading edge the displacementthickness increases from zero and the flow turningoutside of the boundary layer causes compression anda leading-edge shock. The strength of the shockdepends on the incident Mach number and has a strongeffect on the growth of the boundary layer at the leadingedge. Very near the leading edge, the pressure at theedge of the boundary layer is very high compared to thefreestream pressure marking the region of "strong*

interaction. Downstream of the leading edge theinteraction between the inviscid and viscous flow"weakens" and the wall pressure expands to freestream.

From laminar boundary layer theory theappropriate similarity parameter for the shock-boundarylayer interaction is the Lees-Probstein parameter Xgiven by

(33)

where C is the Chapman-Rubesin compressibilityparameter, M_ the freestream Mach number, and Rexthe local Reynolds number. The effects of the couplingbetween the leading-edge shock and the boundarylayer have been predicted by various strong 15,16 andweak17 interaction theories. The complete interactiontheory of Bertram and Blackstock18 unifies the strongand weak interaction theories by bridging their limits ofapplicability. The assumptions used in the developmentof these interaction theories include:

(a) the flow is hypersonic (M«»1)(b) no gas dissociation(c) Prandtl number constant(d) vorticity effects due to the curved leading edge

shock are negligible, and(e) pressure distibution is assumed to vary

exponentially with distance from the leading

The complete theory relates the surface pressureto the interaction parameter X as follows,

P n „, A 3 /Y(Y+1) ry

pT=0-83+TV~2~~ GX

where G is the growth factor defined as,

(34)

(v-H TG = 1.648 •—• {=r̂ + 0.352] (35)

aw

and Taw is the adiabatic wall temperature given by

(36)

for laminar flows. The variation of heat transfercoefficient with surface temperature in the strong-interaction region is tabulated by LJ and Nagamatsu1^ .For Y = 1.4, variation of CH with wall temperature Tw isgiven by

CH[Rex/X]1/2 (37)

where the constant C-| depends on Tw and the ratio ofspecific heats and is obtained from reference 19.

Experimental Measurements

The experimental studies used for validation ofthe finite element approach were conducted in theCalspan 48-inch shock tunnel. These experiments20

have generated heat transfer, surface pressure and skinfriction measurements for hypersonic conditions rangingfrom Mach 11.5 to Mach 18.9.

The Calspan shock tunnel consists of a shocktube driver, followed by a conical expansion section anda contoured nozzle to produce uniform axial flow in thetest section. The tunnel is started by rupturing a doublediaphragm which causes expansion of the air in the highpressure driver section. A normal shock is generatedwith the high temperature and high pressure airbetween the normal shock and the driver-driveninterface. When the shock front strikes the end of thedriven section it leaves a region of stagnant highpressure air. This air is then expanded through acontoured nozzle to desired freestream conditions.

Skin friction transducers, thin-film resistancethermometers and pressure transducers mounted atsuitable locations obtain skin friction, heat transfer, andsurface pressure measurements. Details of theexperimental setup are found in reference 20.

The flow parameters used in the numericalcomputations correspond to the nominal test conditionsfor hypersonic flow over a ramp presented in reference20 and given below

Moo

Re<»TresPd

11.68

2989 R0.3589 psia

The experimental data to be used for comparisons willconsist of measurements obtained for two sets of tunnelruns21 . These measurements correspond to runs 19and 21, and flow conditions for these runs are as follows

Run 19

Moo

TresPd

11.71.734E+05/ft2911 R0.3458 psia

Run 21

Moo =

Reoo =Tres =Pd =

11.641.644E+05/ft3105R0.369 psia

The measurements of runs 19 and 21 yield surfaceheating rates, pressures and skin-friction. Thecoefficients of pressure and skin-friction are obtained bydividing experimental values by the freestream dynamicpressure. The coefficient of heat-transfer is obtainedfrom the heat flux at the wall by the relation,

(38)

where TW, the wall temperature, assumed to be thesame for both runs, is obtained from reference 20 andthe specific heat is assumed constant.

Computational Strategy

The hypersonic flow over a ramp results ininviscid-viscous interactions that have distinctly differentcharacter in various regions of the domain. Near theleading edge, the mesh needs to be well refined anduniform to capture the gradients in both the streamwiseand normal directions. Away from the leading edge, theflow is basically the development of the viscousboundary layer and mesh spacings at the surface in thenormal direction are of concern. At the corner, thecoupling between the inviscid and viscous regionsnecessitate the need for adequate refinement to capturethe separated and attached regions of flow.

The adaptive finite element approach wasimplemented on the NAS Cray-2 supercomputer. Theanalysis is started on an initial mesh and iterations areperformed until the L2 norm of the residuals of theconservation variables dropped over three orders ofmagnitude. The mesh is then refined/coarsened basedon the inviscid and viscous error indicators. On the finalmesh, convergence is monitored by observing changesin the residuals of the conservation variables, the heatflux on the surface and variations in pressure valuesnear the surface at the outflow plane. A local time-stepping procedure that is based on the CFL andviscous stability requirements is used to accelerateconvergence to steady-state and details of thisprocedure are found in reference 13. Typicalcomputational speeds on the scalar code were about2.E-04 CPUs/timestep/node and vectorization effortsare underway to enhance this speed.

An initial analysis of the complete flow domainresulted in inaccurate descriptions of the heating ratesand surface pressures. This analysis indicated the needfor the finite element mesh to have more elements atcritical locations of the flow, such as, at the sharpleading edge and the corner to obtain satisfactorycorrelation with experimental data. The analysisdescribed herein divides the flow domain into threeregions. The three regions are shown in figure 5 andare categorized as : (a) the sharp leading edge section,(b) the flat plate section, and, (c) the ramp section.

Sharp Leading Edge

The finite element mesh that results after multiplerefinements/derefinements depends a greal deal on the"skeleton* mesh used to start the adaptation procedure.A judicious selection of the skeleton mesh models thephysics of the flow better and minimizes the number ofelements used in the analysis. The skeleton mesh usedto model the sharp leading edge is shown in Rg. 6. Theskeleton mesh was designed such that on refinement,aspect ratios of unity exist right at the leading edge andhigher aspect ratios prevail downstream of the leadingedge. The mesh obtained after three refinements on theinitial mesh is shown in Fig. 7(a) and contains 7245nodes and 7235 elements (5781 quads and 1454triangles). Density contours for the leading edge sectionof mesh in figure 7(b) show good definition of theleading edge shock and smooth variation of densitywithin the boundary layer. At the leading edge, theboundary layer displacement effect causes the density

at the wall to increase by an order of magnitude andthen to drop off quite rapidly away from the leading

Details of the finite element mesh very near thesharp leading edge ( box A in figure 7(a)) is shown infigure 8(a) and the density contours on this section of themesh appear in figure 8(b). It is interesting to note the"wiggles" that appear in the contours at x=0.005 wherethe element aspect ratio jumps by a factor of 20 in theflow direction. The effect of this jump is also seen infigure 9 in the distribution of the wall pressures near theleading edge.

A region of the mesh where the element aspectratios change considerably normal to the flow directionis at y = 0.005. This jump is inherited from the macro-elements that constitute the skeleton mesh in figure 6.The effect of the aspect ratio jump in the normal directionis best seen by focusing in on the region at the exitplane (x=0.1) defined by box B in figure 7(a). The finiteelement mesh at the outflow plane is shown in figure10(a) and the distribution of pressure along y at x=0.1 isplotted in figure 10(b). The presence of oscillations nearthe wall and at mesh transitions is very evident in thisdistribution. The bigger "wiggle" occurs where theelement aspect ratio in the normal direction changesabruptly by a factor of ten. The analysis was repeatedwith both the second and fourth order dissipation turnedoff for elements located close to the wall and away fromthe vicinity of the sharp leading edge. This was possibleusing arrays that contain information about elementneighbors. The distribution of pressure at the exit with noexplicit dissipation close to the wall in shown in figure10(c). The pressure jump at the wall is absent andpressure levels within the boundary layer are higher.However, the oscillations caused by the aspect ratiojumps are still present. The distribution of pressure at theexit, shown in figure 11, indicates the presence ofpressure gradients normal to the wall even at regionsclose to the wall. Pressure, being derived from theconservation variables, appears to be more sensitive tomesh transitions than, say, density, the distribution ofwhich is plotted in figure 12, exhibiting a smaller kink aty-0.005.

The surface pressure and heat transfer coefficientobtained from this analysis are compared to predictionsby the interaction theories in figure 13. The resultscompare favorably, especially the heat transfercoefficient. Pressure levels close to the leading edgeare well behaved and do not exhibit the "inverse spike*reported by other researchers^?. The maximumdeviation in pressure levels is seen to be near theleading edge.

It is instructive to take a quick look at the flowphysics close to the leading edge. As seen in Figure 14,very close to the leading edge the flow is near freemolecular flow and can be modelled only on the basis ofkinetic theory. The transitional layer has mixed kineticand continuum properties. The merged layer which liesa little downstream of the transition layer can bedescribed by the Navier-Stokes equations but needsvelocity slip and temperature jump conditions at thewal|23. Downstream of the merged layer, where a

distinct invistid layer develops between the shock andthe viscous layer, is the interaction region. The Navier-Stokes equations with no-slip boundary conditions donot accurately model the physics of the flow up to thisinteraction region. So the accuracy of the finite elementcalculations very close to the leading edge is less thanperfect and discrepancies on comparison with thestrong interaction theories is not unexpected.

Flat Plate Section

Profiles of the conservation variables obtained atthe exit of the leading edge section are used as inflowconditions to the flat plate section.

The skeleton mesh used to model the secondsection of the problem, the flat plate region, is shown infigure 15. After three overall refinements the elements atthe wall were divided into 10 sub-elements according toequation (9). The grid parameter p was set to be 1.02and the subdivision yields elements with aspect ratiosover 200 at the wall. The finite element mesh that resultsafter three refinements on this initial mesh contains 8483nodes and 8593 elements (5151 quads and 3442triangles) and is shown in figure 16(a). Density contourson this mesh in figure 16(b) show the mesh capturingthe leading edge shock and growth of the boundarylayer along the plate. Details of the finite element meshnear the inflow (box A in figure 16(a)) and the densitycontours on this section of the mesh appear in figures17(a) and 17(b). Density profiles at the inflow and theexit of the flat plate section, shown in figure 18, illustrategrowth of the boundary layer and weakening of theleading edge shock as the flow traverses the plate.

A comparison of the density profiles at theoutflow of the leading edge and at inflow to the flat platesection in figure 19 indicates the adequacy of nodaldistribution in the normal direction. Linear interpolationis used at the inflow of the flat plate section which tendsto dampen some of the oscillations present at the exit ofthe leading edge section as illustrated by the pressureprofiles in figure 20. It is to be noted that theseoscillations become fixed at the inflow planes, andhence will influence subsequent analyses.

As with the leading edge section, profiles on theexit plane (x=0.7) of the flat plate section are used todefine the inflow boundary of the ramp section.

Ramp Section

The skeleton mesh used for modelling the rampsection of the flow domain is shown in figure 21. Afterthree overall refinements on this skeleton mesh a layerof structured quadrilateral elements were generated atthe wall. Each element at the wall was divided into tensub-elements with a stretching factor of 1.03. The meshthat results after three refinements on this initial mesh isshown in figure 22(a). Density contours obtained on thismesh appear in figure 22(b) and salient characteristicsof the flow such as the leading edge, induced andresultant shocks are all seen to be well captured. Theability of the adaptive refinement procedure to locateregions of high gradients is exhibited by taking a closerlook at the mesh in the ramp comer (box A in figure

22(a)). The mesh in figure 23(a) and density contourson this mesh in figure 23(b) illustrate capture of theleading edge shock and compression of flow that resultsin the induced shock downstream of this region. The finemesh at the surface is needed to model separation andsubsequent reattachment of the boundary layer in thevicinity of the corner. Explicit dissipation terms forelements that are located close to the surface werezeroed out to ensure accurate computation of surfacequantities.

Details of the flowfield, especially the couplingbetween the inviscid and viscous regions of flow, arebest described by profiles of select variables at specifiedaxial locations. Figure 24 indicates the various x-stationswherein the variation in y direction of quantities such aspressure and temperature are studied. Locationsindicated by A, B, C and D are used to describe flowdetails in the domain up to the comer while locations Eto H detail flow characteristics up the ramp.

The distribution of pressure normal to the surfaceat locations A to D is shown in figure 25. As the flowmoves from the inlet toward the comer, the presence ofthe adverse pressure gradient due to the ramp is clearlyseen. Fluid in the vicinity of the comer cannot overcomethe effects of the pressure gradient as well as the skinfriction at the wall and flow separation results. Theseparated boundary layer becomes a free shear layeroutside of the recirculation region. Pressure profiles infigure 25 and velocity profiles in figure 26 illustrate thiseffect. The boundary layer separation is clearly seen byreversal in flow direction of the u-velocity profiles. Thedistribution of temperature at various x-stations in figure27 shows evidence of increase in boundary layerthickness as the flow approaches the ramp corner.Gradients of the temperature profiles at the surfaceindicate the decrease in heat transfer rates from locationA to location D.

Characteristics of the flow up the ramp aredescribed by the distributions at locations E to H.Pressure profiles at E, F, and G in figure 28 indicate thestrengthening of the induced shock and the change inlocation of the leading edge shock relative to thesurface. The distribution of pressure at exit of the ramp(location H) shows the resultant shock which is derivedfrom the interaction of the leading edge shock with theinduced shock. A weak expansion fan also results fromthis interaction, the effect of which is seen by the smalldrop in surface pressures at location H relative to thatat G. The distribution of density in figure 29 indicates thethinning of the boundary layer as the flow moves up theramp. This reduction in thickness of the boundary layeris also seen in figure 30 which shows u-velocity profilesat various x-stations. The presence of the recirculationis seen by the flow reversal close to the surface atstation E. At the exit from the ramp section, the boundarylayer profile is fully developed and downstream of thisregion the interaction between the inviscid and viscousregions of the flowfield is negligible. Temperatureprofiles at locations up the ramp, shown in figure 31,reflect the thinning' of the boundary layer. The largestgradient in temperature at the surface is seen to be atlocation G indicating the approximate location of peakheating rates.

9

Heat flux, skin friction and pressure coefficientsfor the ramp surface obtained from the finite elementapproach are compared to experimental observationsof Holden21. As mentioned earlier, flow conditions usedin the finite element analysis were conditionsrepresentative of runs 19 and 21 in the Calspan tunnel.Figure 32 compares the finite element predictions forheat transfer coefficient to those obtained from tunnelruns. Heat transfer coefficients computed from the finiteelement scheme compare well with both theexperimental runs for locations upto the corner as wellas halfway up the ramp. Heating rates measured nearthe exit of the ramp for run 19 are about twice that ofboth run 21 and numerical predictions. The finiteelement predictions seem to compare better withexperimental data of run 21 as seen by figure 33. Theheat transfer coefficients match quite well at theseparation and reattachment regions and the values ofthe peak heating rate are in excellent agreement. Themain discrepancies seem to be location of the peak heatrate, which for the analysis is shifted to the left, and asteeper drop in heating rates downstream of thislocation.

Skin friction coefficients calculated from the finiteelement approach are compared to those measured intunnel runs 19 and 21 in figure 34. Levels of the skinfriction coefficient compare well with the experimentaldata for regions unaffected by the recirculation while thesize of the recirculation region is underpredicted by thefinite element approach. Both experimental runsmeasure flow separation at around x=15" while the finiteelement predictions locate separation at x=16". Thelocation of the reattachment point from run 21 is x=20.3"and the finite element procedure locates this point atx=19.3". Unsteadiness of the recirculation bubble, whichcould account for the discrepancies in the finite elementpredictions, was not considered in this analysis.

A comparison of surface pressure coefficientsusing the finite element approach with data fromexperimental runs 19 and 21 appears in figure 35.Pressure levels predicted by the analysis is seen to belower at the inflow and recirculation regions. The rise inpressure up the ramp is much more rapid with the finalsurface pressure falling midway between pressurelevels of runs 19 and 21. The experimental data showssharp peaks in pressure coefficients near the exit whilethe finite element approach indicates a gradualdownward trend. This implies that the interactionbetween the leading edge shock and induced shock isstronger than that predicted by the finite elementapproach. The pressure rise for Mach 11.7 inviscid flowover a 15° ramp is about 17.5 and the finite elementanalysis yields a peak pressure rise of over 22.

Refinement Indicators Revisited

The finite element meshes for analyzing flow overthe ramp were obtained by multiple refinements on theinitial meshes, using the inviscid indicator, with pressureas the key variable, and boundary layer indicator.Proper selections of the key variable and values of aand p", the refinement and derefinement thresholds,

impact heavily on the quality of the resulting finiteelement mesh. The effect of refinement and thresholdparameters on the finite element mesh for the rampsection merits investigation. Meshes that result fromrefinements are evaluated by comparing nodaldistribution in the y-direction for some specified x-station.

Four cases of interest, obtained by varying theadaptation parameters are described below.

Case 1: Key variable - pressure, o=.15, p=.1, viscousrefinement indicator off.

Case 2: Key variable - pressure, o=.15, p=.1, viscouso=.15.

Case 3: Key variable - density, o=.15, |J=.1, viscousrefinement indicator off.

Case 4: Key variable • density, a,(3=.1, viscousrefinement indicator off.

The pressure profile at location A (figure 24) inthe flat portion of the ramp for the initial mesh is seen infigure 36. For both case 1 and case 2, the use ofpressure as the key variable causes the leading edgeshock to be refined. The profile reflecting case 2 alsoshows nodes being added inside the boundary layer.The location of these new nodes are seen in figure 37from the u-velocity profiles. The viscous indicator is seento add elements where the velocity profile is essentiallylinear.

The use of density as the key variable (case 3)indicated by the profile in figure 36 shows refinements atthe shock and the boundary layer. Elements within theboundary layer are refined due to sharp changes in thedensity profile as seen in figure 38. The effect oflowering the refinement threshold is shown in figure 39where the pressure profile that results from case 4 iscompared to that of case 3. More elements at theleading edge shock are refined for case 4 and thepressure profile is very similar to that obtained usingpressure as the key variable and the viscous indicator(case 2) in figure 36.

The profiles in figures 36-39 illustrate theinadequacy of using equation (4) with pressure as thekey variable to capture details of the boundary layer.The use of density as the key variable with appropriatethreshold values causes refinements at the shock aswell as within the boundary layer. The use of the viscousindicator equation (8) leads to excessive refinementswithin the boundary layer, where the velocity profile isessentially linear.

Oscillations in pressure close to the wall isobserved in figures 36 and 39. The pressure levels ofthe added nodes are higher than the values at theadjacent nodes. The inaccuracy is due to pressurebeing derived from conservation variables whileconservation variables at a new node are obtained byaveraging variables at adjacent nodes. A time history ofpressure at such a typical node in figure 40 shows thenodal pressure dropping down to the right levels within

10

200 iterations on this mesh. A procedure based oninterpolating primitive variables at each refinement levelwas also tried. This scheme eliminates inaccuratepressures at new nodes within the boundary layer butshows minimal improvements on convergence rates.Interpolation techniques are being investigated toeliminate this inaccuracy and help increaseconvergence rates.

CONCLUDING REMARKS

An adaptive finite element procedure that usesboth triangles and quadrilateral elements is coupledwith a multi-step Galerkin-Runge Kutta algorithm tomodel laminar hypersonic compressible flows. The finiteelement mesh is adapted in regions of high gradients bythe use of inviscid and viscous refinement indicators.The facility to add in layers of structured quadrilateralelements at the wall for better representation of theboundary layer is included within the framework of theprogram.

The ability of the adaptive finite elementprocedure to capture details of compressible flow withstrong-inviscid viscous interactions is demonstrated bymodelling the Mach 11.7 flow over a ramp. To ensureadequate resolution of complex flow features, the flowdomain is split up into three sub-regions: (a) the sharpleading edge, (b) the flat plate region, and (c) the rampsection.

The accuracy of the finite element solution at thesharp leading edge section is evaluated by comparisonwith predictions of strong interaction theories. Thecomplete theory of Bertram and Blackstock yieldestimates for surface pressure distributions while theheat-transfer coefficient is obtained from the stronginteraction theory of Li and Nagamatsu. Goodcorrelations between the finite element predictions andthe strong interaction theories are obtained for theleading edge section.

The finite element mesh for the rampsection captures the physics of the complicated viscous-inviscid interactions at the corner. Surface coefficients ofheat transfer, skin friction and pressure obtained fromthe finite element calculations compare well with theexperimental measurements of Holden. The effect of theleading edge shock on the surface is seen by the peakin heating rates up near the exit section of the ramp.

Smooth mesh transitions within the boundarylayer is seen to be important to model the flow behaviorwithout any localized "wiggles" in the solution.Elimination of all explicit dissipation within the boundarylayer is essential to obtain accurate predictions of heattransfer, skin friction and pressure coefficients. Therefinement indicator used to key in on the boundarylayer needs to be modified to ignore enrichment atregions where the velocity profiles are linear. The abilityto pick out elements that lie in locations of flow reversalsis also needed for flow domains that include separationand reattachment regions. A refinement procedure thatputs in an excessive number of elements at the surface

is to be avoided since this leads to increasedcomputational effort and slower convergence rates.

The study highlights some of the complexitiesinvolved in modelling compressible viscous flowproblems, especially, high speed flows with strongshock-boundary layer interactions. The physics of theflow as well as surface coefficients obtained from theadaptive finite element approach compare favorablywith theoretical and experimental data. This trend isencouraging and the procedure indicates good potentialfor accurate aerothermal load predictions in the designof high speed vehicles.

REFERENCES

1Lohner, R., Morgan K., and Zienkiewicz, O. C.:"The Solution of Non-Linear Hyperbolic EquationSystems by the Finite Element Method," Int. J. Num.Meth. Fluids Vol. 4 (1984), pp. 1043-1063.

, K. S., Thornton, E. A., Dechaumphai, P., andRamakrishnan, R.: "A New Finite Element Approach forPrediction of Aerothermal Loads-Progress in InviscidFlow Computations," presented at the AIAA 7thComputational Fluids Dynamics Conference, Cincinnati,Ohio, July 15-17, 1985. AIAA Paper No. 85-1533.

^Thornton, E. A., Ramakrishnan, R., andDechaumphai, P.: "A Finite Element Approach forSolution of the 3D Euler Equations," presented at AIAA24th Aerospace Sciences Meeting, Reno, Nevada,January 6-8, 1 986, AIAA-86-01 06.

4 Dechaumphai, P., Thornton, E.A. and Wieting,A.R.:"Flow-Thermal-Structural Study of AerodynamicallyHeated Leading Edges," Presented at theAIAA/ASME/ASCE/AHS 29th Structures, StructuralDynamics and Materials Conference, Williamsburg,Virginia, April 18-20, 1988.

^Zienkiewicz, O. C., Lohner, R., and Morgan, K.:"High Speed Inviscid Compressible Flow by the FiniteElement Method," presented at the MAFELAP-VConference, May 1-4, Brunei, UK.

6Stewart, J.R., Thareja, R.R., Wieting, A.R. andMorgan, K.: "Application of Finite Element andRemeshing Technique to Shock Interference on aCylindrical Leading Edge", Presented at the AIAA 26thAerospace Sciences Meeting, Jan 11-14, 1988, Reno,Nevada. AIAA-88-0368.

7Lohner, R.: "The Efficient Simulation of StronglyUnsteady Flows by the Finite Element Method,"Presented at the AIAA 25th Aerospaces SciencesMeeting, Reno, Nevada, January 12-15, 1987, AIAA -87-0555.

, J. T., Demkowicz L., Strouboulis, T., andDevloo, P. : "Adaptive Methods for Problems in Solidand Fluid Mechanics," in I. Babuska , O. C. Zienkiewicz,J. P.de S. R. Gago, and A. de Oliveira , (Eds.),

11

Adaptive Methods and Error Refinement in FiniteElement Computation. John Wiley and Sons. Ltd..London 1986.

^Shapiro, R. A., and Murman, E. M.: "Cartesian GridFinite Element Solutions to the Euler Equations,*presented at the AIAA 25th Aerospace SciencesMeeting, Reno, Nevada, January 12-15, 1987, AIAA-87-0559.

10peraire, J. , Vahdati, M. , Morgan, K. , andZienkiewicz O. C. : "Adaptive Remeshing forCompressible Flow Computations," Journal ofComputational Physics, 72, 449-466, 1987.

11Thareja, R.R., Stewart, J.R., Hassan, O., Morgan,K., and Peraire, J.: "A Point Implicit Unstructured GridSolver for the Euler and Navier-Stokes Equations,"Presented at the AIAA 26th Aerospace SciencesMeeting, Jan 11-14, 1988, Reno, Nevada. AIAA-88-0036.

12Ramakrishnan, R., Bey, K.S., and Thornton, E.A.:"An Adaptive Quadrilateral and Triangular FiniteElement Scheme for Compressible Flows," Presented atthe AIAA 26th Aerospace Sciences Meeting, Jan 11-14,1988, Reno, Nevada. AIAA-88-0033.

13Thomton, E.A., Dechaumphai, P., and Vemaganti,G.: "A Finite Element Approach for Prediction ofAerothermal Loads," Presented at the AIAA/ASME 4thFluid Mechanics, Plasma Dynamics and LasersConference, May 12-14, 1986, Atlanta, GA. AIAA-86-1050.

14Roberts, G.O.: "Computational Methods forBoundary Layer Problems," Proceedings of the SecondInternational Conference in Numerical Methods in FluidDynamics, Lecture Notes in Physics, Vol. 8, Springer-Verlag. NY. pp 171-177.

15Li, T.Y. and Nagamatsu, H.T., Journal ofAeronautical Sciences, Vol. 20, No. 5, 1953, pp. 345-355.

16|_ees, L. and Probstein, R.F.: "Hypersonic ViscousFlow over a Flat Plate". Report No. 195 (Contract AF33(038)-250), Aero. Eng. Lab., Princeton Univ., April,1952.

17Hayes, W.D. and Probstein, R.F.: "HypersonicFlow Theory", pp. 341-353, Academic Press, Inc., NewYork, 1959.

18Bertram, M.H. and Blackstock, T.A.: "SomeSimple Solutions to the Problem of PredictingBoundary-Layer Self-Induced Pressure, NASA TN D-798. April 1961.

I^Dorrance, W.H.: "Viscous Hypersonic Flow." pp154-157, McGraw-Hill Book Company, 1962.

2°Holden. M.S.: "A Study of Flow Separation inRegions of Shock Wave-Boundary Layer Interaction inHypersonic Flow," Presented at the AIAA 11th Fluid andPlasma Dynamics Conference, Seattle, Wash., July 10-12. 1978.

21Calspan Corporation, "Laminar Flow Data Base,"August 26. 1987.

22Ray. R., Erdos, J. and Pulsonetti. M.V.:"Hypersonic Laminar Strong Interaction Theory andExperiment Revisited Using a Navier-Stokes Code,"Presented at the 19th Fluid Dynamics, PlasmaDynamics and Lasers Conference, June 8-10, 1987,Honolulu, Hawaii.

23Tannehill, J.C., Mohling, R.A-, and Rakich. J.V.:"Numerical Computation of the Hypersonic RarefiedFlow Near the Sharp Leading Edge of a Flat Plate,"Presented at the AIAA 11th Aerospaces SciencesMeeting, Washington, D.C., January 10-12.1973.

(a) Triangular enrichment

Hinging nod*

(b) Quad enrichment

(c) Quad enrichment with triangulartransition

Fig. 1 Mesh Enrichment using quadrilateral andtriangular elements.

12

f f / S S //////

Fig. 2 Velocity profile at wall and unit vector definingchange of velocity.

-Hoi— ,i-H

(a) Original Mesh

Od 03 CO

Cl

C3

C2

(b) Refinement of elements B, C, and D

7CD2

(c) Derefinement of element C

Fig. 3 Example of adaptation procedure usingtriangular transition elements.

Resultantshock -

Flow

3M. 11.6

InducedCompression shock -

Leading edge fan"shock- Slip surface

Expansion fan

Reattachmentpoint

Boundary/ '-Separation ^ Redrculationlayer -> point

Fig. 4 Hypersonic flow over a 15° ramp.

Fig. 5 Subdivision of hypersonic flow domain.

.005

Sharpleading edge

Plate

Rg. 6 Skeleton mesh for sharp leading edge.

(a) Rnite element mesh

Fig. 7 Finite element mesh and density contoursfor sharp leading edge.

13

(b) Density contours

Rg. 7 Finite element mesh and density contoursfor sharp leading edge.

Aspectratio

effect

.004 .008 .012 .016 .020

x - . O

(a) Rnite element mesh

Aspect ratiojump

Rg. 9 Pressure distribution in the vicinity of theleading edge.

-Aspectrstioeffect

X..1 .011 ~OM .013 .014.011 .012 .013 .014Pressure Pressure

(a) (b) (c)

Rg. 10 Rnite element mesh and pressuredistribution at the exit plane of sharpleading edge.

(b) Density contours

Rg. 8 Rnite element mesh in the vicinity of theleading edge.

.05

.04

.03

.02

.01

.005 .010 .015 .010Pressure

.025

Rg. 11 Pressure distribution at exit plane of theleading edge section.

14

.000.3 .4

Density.5

M

SHOCK WAVE

INVISCID REGION

STRONG WEAKINTERACTION INTERACTION

•WALL SLIP- •NO WALL SLIP-

_OC

Bg. 12 Density distribution at exit plane of theleading edge section.

Fig. 14 Hypersonic flow detail at sharp leadingedge.

30

25

20

15

10

5

0

Rnite element

'H

.10

.08

.06

.04

.02

F

Strong interactiontheory, Eq. 34

.02 04 .06X

.08 .10

(a) Surface pressure ratio

Rnite elementStrong interaction

theory, Eq. 37

.02 04 .06X

.08 .10

(b) Heat transfer coefficient

Fig. 13 Comparison of pressure and heat transfercoefficients for the sharp leading edge.

FU t plate seel

>tiii}>»

ion

jjf"""

i

.5

TosjPlate

Fig. 15 Skeleton mesh for flat plate section.

(a) Finite element mesh

Fig. 16 Finite element mesh and densitycontours for flat plate section.

15

(b) Density contours

Fig. 16 Finite element mesh and densitycontours for flat plate section.

x = .11 x = .15

(a) Finite element mesh at inflow

(b) Density contours at inflow section

Fig. 17 Detailed mesh at inflow for flat platesection.

.20

.16

.12

.08

.04

— Exrt of flat plate— Inlet of flat plate

0 .5 1.0 1.5 2.0 2.5 3.0Density

Fig. 18 Comparison of inlet and exit profiles offlat plate section.

.05

.04

.03

.02

.01

Exit leading edgeInlet flat plate

0 .5 1.0 1.5 2.0 2.5 3.0Density

Fig. 19 Density profiles at exit of the leadingedge and inflow to the flat plate section.

•010_

•QQ8

.006

.004

.002

.000

Inlet flat plateExit leading edge

.010 .011 .012 .013Pressure

.014 .015

Fig. 20 Effect of interpolation procedure onpressure profiles at inflow to flat platesection.

16

x-0.7 x- 1.45

Fig. 21 Skeleton mesh for ramp section. x- 1.3

(a) Finite element mesh

x - 1.65

(a) Finite element mesh

(b) Density contours

Fig. 23 Mesh detail at comer of ramp.

(b) Density contours

Fig. 22 Finite element mesh and density contours forramp section Fig. 24 Locations of profile stations along the ramp.

17

PressurePressure

Fig. 25 Pressure distribution at locations on flat portionof ramp.

Fig. 28 Pressure profiles at .locations up the ramp.

.02

Density

Fig. 26 Velocity profiles at locations on flat portion oframp.

Fig. 29 Density profiles at locations up the ramp.

.04 .OB .12

Temperature

-.1 .1 .3 .a .7 .9

U-Velocity

Fig. 27 Temperature distribution at locations on flatportion of ramp.

Fig. 30 Velocity profiles at locations up the ramp.

18

Cf

-.002

* Run 19o Run 21

— Finite Element

1.2

s

Fig. 31 Temperature profiles at loactions up the ramp. Fig. 34 Comparison of skin friction coefficient on rampsurface.

o Run 19* Run 21— Finite Element

CH

1.2

S

Cp

* Run 19o Run 21- Finite Element

InviscidSolution

1.2

S

Fig. 32 Comparison of heat transfer coefficients on theramp surface.

Fig. 35 Comparison of surface pressure coefficients onramp surface.

* Run 21— Finite Element

CH

1.2

S

Initial ProfileCase 1Case 2Case 3

I I I' .

Pressure

Fig. 33 Comparison of heat transfer coefficients on the Fig. 36 Pressure profiles at location A showing effects oframp surface. refinement.

19

PRESSURE

Initial ProfileCase 1Case 2

Initial ProfileCase 3Case 4

.00

U-Velocity Pressure

Fig. 37 Velocity profiles at location A on ramp. Fig. 39 Pressure profiles on ramp showing effects ofvarying threshold values.

Initial ProfileCase 1Case 2Case3

IinM

DensitySOO. 1000.

Iterationsisoo.

Fig. 38 Density profiles on ramp showing effects of Fig. 40 Temporal history of pressure at typical new noderefinement. within the boundary.

20