[American Institute of Aeronautics and Astronautics 49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition - Orlando, Florida ()] 49th AIAA

American Institute of Aeronautics and Astronautics

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Validation and Verification of the Navier-Stokes Solutions Obtained from the Applications of the

Integro-differential Scheme

Frederick Ferguson 1, Mookesh Dhanasar2, and Nasstaja Dasque3 Center for Aerospace Research, NCAT, Greensboro, NC 27411, USA

The goal of this research effort is to improve the efficiencies of Computational Fluid Dynamic tools by focusing on the development of a robust, efficient, and accurate numerical framework. What is most desirable is a framework that is capable of solving a variety of complex fluid dynamics problems over a wide range of Reynolds and Mach numbers, and one with the potential to overcome several limitations of well-established schemes. Preferable, is a computational framework that might minimize the dispersive and dissipative tendencies of traditional numerical schemes. To this end, a new scheme, termed the ‘Integro-Differential Scheme’ (IDS) and sometimes called the ‘Method of Consistent Averages’ (MCA), Ref [1-2], is proposed. The MCA numerical procedure developed herein is based on two fundamental types of control volumes; namely, spatial cells and temporal cells. A typical spatial cell is developed from eight neighboring nodes, whereas, a typical temporal cell is developed from the centroid of the eight neighboring spatial cells. Further, each cell in this analysis, acts as a control volume on which the Navier-Stokes equations are integrated. The final solution at a single node is then developed from a system of carefully crafted control volumes. This paper describes the results obtained from applying the IDS method to a set of selected 2D fundamental fluid dynamic problems; namely, (1) the Hypersonic Leading Edge problem, (2) the incompressible Lid Driven Cavity problem, (3) the Shock Boundary Layer Interaction Problem and (3) the Isolator Pseudo-Shock Train problem.

Nomenclature A = amplitude of oscillation a = cylinder diameter Cp = pressure coefficient Cx = force coefficient in the x direction Cy = force coefficient in the y direction c = chord dt = time step Fx = X component of the resultant pressure force acting on the vehicle Fy = Y component of the resultant pressure force acting on the vehicle f, g = generic functions h = height i = time index during navigation j = waypoint index K = trailing-edge (TE) non-dimensional angular deflection rate

1 Professor, Mechanical Engineering Department, NCAT, and AIAA Associate Fellow. 2 Research Associate, Mechanical Engineering Department, NCAT, and AIAA Member. 3 Graduate Student, Mechanical Engineering Department, NCAT, and AIAA Member.

49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition4 - 7 January 2011, Orlando, Florida

AIAA 2011-1300

Copyright © 2011 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.


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I. Introduction historic review of the computer industry indicated that it has completed four generations and is now entering its fifth generation1-6. Engineers have labeled computers that use the vacuum tubes as the first generation

computers, and those that used the transistors and diodes as the second generation of computers. The integrated circuits (ICs) ushered in the third generation computers. Currently we are enjoying the benefits of the fourth generation computers that were made possible thanks to the development of micro-processors1-6.

Arguably, the fifth generation of computers is already with us. The development of integrated ‘software and multiprocessors’, of the type facilitated by field programmable gate arrays (FPGA) and others, are ushering a new generation of ‘Peta-scale’ computers. In principle, computer engineers have developed a common consensus on the path toward the fifth generation of computer based systems. This path involves the global unification of software components that interacts with a series of hardware coprocessors. The fifth generation of computers is expected to deliver performances in the order of petabytes (storage capacity in the order of 10 million or more gigabytes) and pataflops (one quadrillion floating point operations per second). In the US, the National Science Foundation and DARPA have initiated funding for the development of such computers. In fact, DARPA has contracted with IBM through the PERCS (Productive, Easy-to-Use, and Reliable Computer System) program. Other countries, such as, China, Germany and Japan, have similar programs. The immediate computational focus of the ‘Peta-scale’ computers under development will be on weather and climate simulations, nuclear and quantum chemistry simulations, and cosmology and fusion science simulations1-6. The motivation of the research program described herein is targeted towards the development of a Computational Fluid Dynamic tool that seeks to maximize the benefits of future ‘Peta-scale’ computers2.

A: Historic Perspective of CFD as an industrial design tool A historic view of the Fluid Dynamic industry, especially, as it relates to the aerospace industry, indicates that

there are at least three major breakthroughs in CFD. The first breakthrough occurred in the 1950’s and 1960’s. During this period, numerical methods and grid generation techniques were merged into the formation of Computational Fluid Dynamics (CFD) as a special branch of fluid mechanics. However, in the 1950s and 1960s CFD did not play a dominant role, mainly due to the lack of computational facilities. Nevertheless, numerical methods were developed and applied to 2D flows. In those days, the results were always supported by experimental data, thus validating the importance of CFD1-6.

A second breakthrough in CFD occurred in the 1970s, when the science of orthogonal surface-fitted structured grids was introduced. During that time, the storage and speed of selected and limited digital computers were sufficient to conduct CFD studies on flow problems of practical interest. Using structured surface-fitted grids, the computations of flow over airfoils and wings became accurate, affordable and efficient. This breakthrough led to remarkable improvements in the design and performance of fixed wing aircrafts.

The third breakthrough in CFD occurred in the 1980s and 1990s. During this time, the computer had fully penetrated the industrial market and was well on its way to making it into every home. Additionally, there were remarkable improvements in both computational speed and storage. At this time, numerical algorithms and grid generation techniques were expanded and included unstructured grids on realistic aircraft configurations. The successful aerodynamic analysis of the full size aircraft configurations using CFD was demonstrated3. It was also in the 1990s that CFD analysis became integrated into other industries, such as, the automobile, civil and environmental industries.

CFD as a design tool is playing a leading role wherever a flow field is being analyzed. In fact, CFD has significantly influenced the way engineers analyze problems and conduct design. However, not withstanding these successes, there are still great CFD challenges remaining today1-2.

B: The Challenges of the CFD Industry Typically, any problem facing the CFD engineer can be solved through the integrated use of the following three

elements7-12: (i) the available or intended computational hardware, (ii) an appropriate form of the conservation laws and their constitutive relations, and (iii) a specified set of computational grid points.

As such, to understand the challenges facing the CFD industry, one must put these three elements in proper perspective.

A


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The computational hardware of interest to this research was described in the introduction section of this paper. The motivation of the research program described herein is geared towards the development of a Computational Fluid Dynamic tool that seeks to maximize the benefits of future ‘Peta-scale’ computers.

The conservation laws and their constitutive relations are usually specified by the CFD designer. In most instances, these equations are chosen based on the capability of the available computational hardware and in the case of industrial software, the available turbulence model. The appropriate form of the conservation laws and their constitutive relations of interest to this research is the Navier-Stokes equation with the Reynolds Stress Model. Designers are currently focusing on problems where turbulence has the dominant effect and problems where the traditional two-equation models are no longer adequate. Clearly no proper evaluation of the merits of different turbulence models can be made unless the discretisation error of the numerical algorithm is known. As such, grid sensitivity studies are crucial for all turbulence model computations.

C: Computational Grid Points The successful analysis of any CFD design problem can be traced back to the successful development of its grid

structure. As such, in order to appreciate the challenges facing the CFD Industry one must take a critical look at the grid generation techniques available. Typically, the grid generation procedure is left up to the CFD designer, who has one of two choices, structured or unstructured grids methods.

C1: Structured Grid Methods Structured grid methods take their name from the fact that the grid is laid out in blocks of regular repeating

pattern. These grids utilize quadrilateral elements in 2D and hexahedral elements in 3D. Although the element topology is fixed, the grid can be shaped to be body fitted through stretching and twisting of the block. In reality, structured grid tools utilize sophisticated elliptic equations to automatically optimize the shape of the mesh for orthogonality and uniformity. Structured grids can be arranged in multiple blocks, with and without overlapping. Refer to Figure 1. Structured grids enjoy a considerable advantage over its unstructured counterpart, in that they allow a high degree of control. In addition, hexahedral and quadrilateral elements, tolerates a high degree of skewness and stretching without significantly affecting the solution accuracy in the case of well behaved flowfields. The major drawback of structured block grids is the time and expertise required to lay out an optimal block structure for industrial size model. Grid generation times are usually measured in weeks if not months.

Figure 1: Structured Overlapping Grids [3]. Figure 2: Unstructured Grids [3].

C2: Unstructured Grid Methods Unstructured grid methods utilize an arbitrary collection of elements to fill the domain. Due to the fact that the

arrangement of the elements has no discernible pattern, the mesh is called unstructured. Refer to Figure 2. These types of grids typically utilize triangles in 2D and tetrahedra in 3D. As with structured grids, the elements can be stretched and twisted to fit the domain. These methods are usually automated. Given a good CAD model, a good designer can automatically place triangles on arbitrary surfaces and tetrahedra in the arbitrary volume with very little


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effort. The major advantage of unstructured grid methods is that they enable the solution of very large and detailed problems in a relatively short period of time. Compared to structured grids method, unstructured grid generation times are usually measured in minutes or hours.

However, the major drawback of unstructured grids is the lack of user control when laying out the mesh. Typically user involvement is limited to the boundaries. In addition, triangle and tetrahedral elements are resistant to stretching and twisting, therefore, the resulting grids are isotropic, with all elements having roughly the same size and shape. This is a major problem when trying to refine the grid in a local area, often the entire grid must be made much finer in order to get the point densities required locally. Another drawback of this method is its reliance on geometric input or CAD data. Typically, most meshing failures are due to some minuscule error in the geometric input. Unstructured flow solvers typically require more memory and have longer execution times than structured grid solvers on a similar mesh.

D: Potential Benefits of Cartesian Grid Flow Solvers The pros and cons of the state-of-the-art in grid generation science outlined in section C, and the potential

computational capabilities of the next generation computers outlined in section A provide a significant opportunity for the CFD industry13-16. Opportunities exist to overcome the existing challenges of grid generation and to aggressively tackle the direct simulation of turbulent dominated flows over complex configurations13-16.

Potentially, significant reductions in aircraft design cycle times through the automated generation of structured grids with optimum surface resolution can be achieved. These reductions will not come from increased reliance on user interactive methods, but instead from methods that fully automates and incorporates themselves into ‘black boxes’. In comparison with unstructured grid methods, 3D Cartesian grid approaches are still in its infancy. However, the recent development of automated Cartesian grid generation techniques are quite promising1-2. Our research is targeted at furthering the development of Cartesian methods so that they can become key elements of a completely automatic and coupled grid generation and flow solution procedure1-2.

Cartesian approaches are of course beset with their own unique difficulties. The most challenging is the removal of the body-fitted grid constraint. This allows the Cartesian hexahedra used to discretize the flowfield and to intersect the surface in an arbitrary manner. Successful research into the development of robust procedures for the efficient creation and distribution of the hexahedra will produce an automatic procedure for Cartesian grid generation. Another potential benefit of this research is the ability to solve complex turbulent flow fields with RSM over industrial size configurations afforded by the simplicity of Cartesian grids.

II. Current Research Focus The immediate goal of this research is focused on the development of a robust, efficient, and accurate numerical

framework that is capable of solving a variety of complex fluid dynamics problems on Cartesian grids1-2. The IDS scheme is built with extensive physics considerations and has the following features: i. The numerical scheme is based on the solution of the integral form of the Navier-Stokes equation. This

approach focuses on the benefits of the traditional finite volume and finite difference schemes, and therefore guarantees the conservation properties throughout the domain by the first and the formulation simplicity by the latter.

ii. The Cartesian grid generation procedure is used to develop spacial and temporal control volumes upon which the integral form of the physical conservation laws are applied. As such, the scheme has the potential to satisfy the physical realities of fluid fluxes for both time and space.

iii. Accounting of the mass, momentums, and energy fluxes (both within the control volume and through its surfaces) is conducted with the aid of the mean value theorem, rather than the traditional extrapolation or interpolation of the node’s information from neighboring cells.

iv. The accuracy and efficiency of the scheme is demonstrated on flows with rectangular boundries.

A: The Governing Equations The equations that govern fluid flows and the associated heat transfer are the continuity, momentum and energy

equations. These equations were independently constructed by Navier (1827) and Stokes (1845), and are referred to as the Navier-Stokes equations. In this research, the integral forms of the Navier-Stokes equations (1–3) are of paramount importance. The continuity, momentum and energy equations are listed as follows:


5

v s

0sdVdvt

Vd (1)

sdsPdVsdVdvVt ssv

Pt

dVV ˆ.

(2)

ssssv

sdqsdVsdVPsdVEEdvt

.̂.. V.̂̂EE (3)

In Equations (1) the symbols, tv,,, , represent the density, the volume of a control fluid element, and time, respectively. In addition, the symbols, V , sd and q , in equations (1), (2) and (3) represent the fluid velocity, the surface of the control volume and the local heat transfer rate. In this research, fluid velocity and the surface element are described through the use of vector quantities as follows:

kwjviuV u (4)

kdxdyjdxdzidydzsd d (5)

and

kqjqiqq zyx qqq (6)

In equations (2) and (3), the symbol, P, represents the pressure and the symbol, ,ˆ,ˆ represents a symmetric tensor that defines the various components of the local viscous stresses. This symmetric tensor is described by the equation:

zzzyzx

yzyyyx

xzxyxx

zzzyz

yyyy

xxyx

ˆ (7)

where the symbols of the six independent components, zyzxyxyyxyxx zyzxyyxyx ,,,,, and zzz , are the local shear stress that were defined in equation (7) as follows:

xu

Vxx xu

Vxx 2. 32 (8)

xv

yu

yxxy yxy (11)

yv

Vyy yv

Vyy 2. 32 (9)

xw

zu

zxxz zx (12)

zw

Vzz zw

Vzz 2.32 (10)

zv

yw

zyyz zy (13)

The symbols, yx qq , and zq , in equation (3) represent the components of the heat flux vector in the x- , y-, and z-

directions, respectively. These components are defined by Fourier’s law, and expressed mathematically as,


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zT

kq

yT

kq

xT

kq

z

y

x

zT

k

yT

k

xT

k

(14)

The symbols, P and E, in equations (2) and (3) are defined as follows:

2

222 wvuTCE

RTP

vwvu

C

R (15)

where R is the universal gas constant. The symbols, and k, represent the viscous and thermal properties of the fluid of interest. In this analysis, the viscosity of the fluid is evaluated through the use of Sutherland’s law7-8,

110T110T

TT

23

11

3

TTTTT TT

/

(16)

where the symbols, and T , represent the freestream dynamic viscosity and temperature. In the case of 3D aerodynamic analysis, the Navier-Stokes equations (1)–(3) defined above can be treated as a closed system of five equations relative to five unknowns. In this analysis, the unknowns are the following five primitive flow field variables: Twvu ,,,,, . The immediate goal of this research is to develop an explicit method that solves the Navier-Stokes system of equation on a set of Cartesian grids in which arbitrary objects are immersed.

III. The Integro-Differential Scheme (IDS) A typical fluid flow is represented by a rectangular domain1-2. This spatial computational domain is further

divided into a collection of elementry cells, herein called ‘spatial’ cells. These cells are chosen as infinitesimal rectangular prisms, with unit normal, ñ, in the x, y, and z directions. The dimensions of each side of a cell are defined by dx, dy, and dz, respectively. Refer to Figure 3. Further, a given cell is defined locally by six independent surfaces, and each surface defined by four points or nodes in a given plane. Additionally, plus and minus notations are use to define the unit normal, ñ, with respects to each surface. Next, in the case of a 2D domain, (the 3rd dimension is held fixed) each surface of a given cell is defined by four nodes; namely, nodes-1, nodes-2, nodes-3 and nodes-4. Figure 3 illustrates the plus and minus notations for the surfaces with normal to the z-direction. It is of interest to note that the use of the object oriented programming concept makes it very convenient to use identical surface objects in the x and y directions.

Analogous to ‘spatial’ cells, the concept of ‘temporal’ cells are also introduced. The ‘temporal’ cells are defined as rectangular prisms formed from the centroids of eight neighboring ‘spatial’ cells. Finally, a control fluid volume is defined, as a collection of eight ‘spatial’ cells and one temporal cell. A typical control volume is illustrated in Figure 4. Each term in the Navier-Stokes equations (1–3) are applied systematically to each spatial cell. The mean value theorem is invoked and a set of algebraic equations representing the rate of change of mass, momentum, and energy associated with each spatial cell is derived. However, the rates of change of the time-fluxes are not associated with any grid point, but with the ‘spatial’ cell. When the spatial cells are pieced together to form a temporal cell within the control volume, the arithmetic average of the rates of change within the temporal cell then defines rates of change at the ijk-point of interest.

It is of interest to note that the plus and minus surfaces in each direction is adequate to evaluate the invicid fluxes. However, two additional and adjacent surfaces in each direction are needed for evaluating the viscous terms. In this analysis, these surfaces are denoted as the plus-plus and minus-minus surfaces, and are treated in the manner described earlier.


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Figure 3: Spatial Cell with Notation at Surface Nodes. Figure 4: Illustration of Control Volume.

IV. Preliminary Results As described earlier in this paper, the immediate goal of this research effort is to develop a computational tool

that accurately solves the Navier-Stokes equations in the Cartesian system of coordinates. In this section, the following three standard CFD problems are solved to demonstrate the validity of the new computational tool: i. Flow over a flat plate at Subsonic, Transonic and Hypersonic Conditions ii. Incompressible and Supersonic Lid Driven Cavity flow Problem iii. Shock Train Problem

The solution procedure adopted during the numerical simulation of these problems is as follows: i. The physical domain of the problem is defined in Cartesian Coordinates. ii. The freestream parameters of interest ( , M, Re, and Pr) are mapped to fit the Navier-Stokes equations

requirements and the physical domain described in Section I. iii. The boundary conditions are assigned in accordance with Method of Consistent Averages methodology. iv. The solution grid and initial solution are auto-generated. v. The solution data is extracted from the 3D domain through planes and lines at the locations of interest.

A: Hypersonic Flow over a Flat Plate Consider a developing boundary layer on a flat-plate under supersonic conditions, as illustrated in Figure 5. An

oblique shock wave develops at the leading edge of the flat plate due to the viscous boundary layer effects. Refer to Figure 6. The occurrence of the boundary layer often provokes dramatic changes in the flow field features, both qualitatively and quantitatively. Important consequences of this occurrence are the increase in the skin friction coefficient and the shear stresses inside the boundary layer region. Moreover, dissipation of kinetic energy within the boundary layer can cause high flow-field temperatures and thus high heat-transfer rates. The supersonic flow over a flat plate is a classical fluid dynamic problem, and it has received considerable attention from many researchers; including Anderson7, Akwaboa17, MacCormack18 and Rasmussen19.

The results of this study are illustrated in Figures 6–10. Figure 6 illustrates a contour plot of the pressure distribution over the plate. Evidence of a shock wave that developed at the leading edge of the plate as a result of the boundary layer can be clearly detected in Figure 6. In an effort to validate the numerical solution, a grid independence study was carried out on four sets of Cartesian grids; namely, a 101 by 101, 201, 201, 301 by 301 and a 401 by 401 rectangular grid systems, respectively. The residual as a function of the number iterations are plotted in Figure 8, for each grid size. In each case, the results were consistent, the residual error, exponentially decreased with respects to time. The u-velocity, temperature and pressure data are also presented in Figures 8–10. Figure 10 indicates the accuracy of this method as compared to the available experimental data, Ref5.


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Figure 5: Hypersonic Boundary Layer7 Figure 6: Pressure distribution in Boundary Layer

Figure 7: The exit Temperature in Boundary Layer Figure 8: Pressure distribution in Boundary Layer

Figure 9: The exit Temperature in Boundary Layer Figure 10: Pressure distribution in Boundary Layer

Shock wave

Iterations

Res

idua

l

1 5001 10001

10-14

10-12

10-10

10-8

10-6

10-4

10-2

201 by 201 Grid

401 by 401 Grid

101 by 101 Grid 301 by 301 Grid

Non-Dimensional uVelocity

Y

0 0.25 0.5 0.75 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

101 by 101 Grid

401 by 401 Grid

Non-Dimensional Density

Y

0.9 1 1.1 1.2 1.3 1.40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

401 by 401 Grid

101 by 101 Grid201 by 201 Grid301 by 301 Grid

X

Non

-Dim

ensi

onal

Pres

sure

0 0.5 11

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

101 by 101 Grid

401 by 401 Grid

201 by 201 Grid


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B: The Lid Driven Cavity Problem Consider a 3D incompressible flow field that is confined to a cavity, where the motion in the cavity is generated

by the sliding motion of the fluid at a plane of one of its surfaces. This moving fluid surface generates vorticity which diffuses inside the cavity. The diffusion effect now becomes the dominant mechanism driving the flow. As part of the validation studies of the Navier-Stokes code developed herein, the lid driven cavity problem was solved. Figures 11 and 12 illustrate the results obtained from this IDS procedure, and compare them against the experimental findings of both Choi & Merkle20 and Vigneron et.al21.

Figure 11: The x Component Velocity Profile along

the Vertical Centerline of the Cavity Figure 12: The y Component Velocity Profile along

the Horizontal Centerline of the Cavity

The data presented in Figures 11 and 12 are the steady state solution for the x-component of the velocity on the vertical centerline and the y-component of the velocity on the horizontal centerline of the cavity. It is of interest to note that incompressible solutions are obtained without introducing any modifications to the IDS algorithm. Another perspective of the cavity solution in the form of streamlines is presented in Figures 13. Again, the IDS solution is compared to the Vigneron et.al21 solution, see Figure 14, for the cavity problem under the same operating conditions. As observed in these Figures 13 and 14, the IDS solution shows very good agreement with the results of Vigneron et al21.

Figure 13: Cavity Streamlines plot of the IDS

Solution Figure 14: Cavity Streamlines plot Vigneron Solution

0.00.10.20.30.40.50.60.70.80.91.0

-0.3 0.0 0.3 0.5 0.8 1.0

IDS results Choi & Merkle (1995) Vigneron et al. (2008)

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.00 0.20 0.40 0.60 0.80 1.00

IDS Results Choi & Merkle (1995) Vigneron et al.


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C: The Shock-Boundary Layer Interaction Problem Consider the Shock Boundary Layer Interaction problem22, which is one of the standard test problems from

Navier-Stokes solvers. Further, consider the scenario where the shock wave is strong and the incident angle of the shock wave is also large. In this case, separation occurs at the impingement point. Usually, to resolve the details of this flow field phenomenon, clustered grids are used in the separation region. However, in the case of the IDS procedure, no such clustering techniques are used. A rectangular set of grid points; 301 by 601, (approximately 180,901 cells) were used in this study.

In this analysis, the freestream Mach number is set to 3.0, the Reynolds number to 106, the Prandtl number to 0.71 and shock angle, , is set to atan(5.0/7.0) (approximately 40.9256 deg). The problem is solved on a set of uniform grids. The results of this study are illustrated in Figures 15–18. Figures 15 and 16 presents the distribution of the x- and y-components of the velocity, with over plots of the velocity field. Figure 17 highlights the normal pressure distribution at locations of 20%, 25%, 50% and 100% along the length of the plate. On the other hand, Figure 18 shows the distribution of the pressure and the shear stress along the length of the plate. These results represent the physical behavior of the flow field.

Figure 15: The x Component Velocity Profile Figure 16: The y Component Velocity Profile

Figure 17: The SBLI Normal Pressure Profile Figure 18: Wall Pressure and Shear Stress Profile

Non-Dim X

Non

-Dim

Y

0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1u-Velocity

0.92780.85060.81720.81050.80490.79420.73020.51440.29850.0827

-0.0554

Non-Dim X

Non

-Dim

Y

0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1v-Velocity

0.11300.07320.0333

-0.0066-0.0465-0.0863-0.1262-0.1661-0.2060-0.2458


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D: The Jet and Cross Flow Interaction Problem The flow field resulting from a gas that is injected traversely into a supersonic crossflow is encountered in many aerospace applications, such as, the thrust vectoring of rocket motors, supersonic combustion in scramjets, and the cooling of gas turbine blades. The flowfield interaction of the jet and the crossflow results in shock waves, boundary layer separation and recirculation zones23, as illustrated in Figure 19. Typically, a bow show forms ahead of the injection which interacts with the boundary layer to result in a separation bubble. In addition, the jet core flow penetrates the boundary layer and terminates in either a normal shock wave or a Mach disk. Directly downstream of the jet plume a second separation zone develops, refer to Figure 20. In 3D flowfields, horseshoe vortexes are also formed23.

Figure 19: The Jet-Cross Flow Problem23 Figure 20: Cross flow Velocity and Pressure Profiles

In this analysis, the jet and cross flow interaction problem23 is solved for the conditions prescribed in Ref. 24. For this study, a rectangular set of grids, 401 by 401, was used. The preliminary results obtained from the use of IDS procedure are illustrated in Figures 20–22. Figure 20 illustrates a close up of the flow field at the jet injection zone, whereas, Figure 21 illustrates a stream trace in the entire domain. Figure 22 demonstrates the validity of the numerical calculations, whereby the pressures at key points in the flow field are recovered.

Figure 21: IDS Jet-Cross Flow Streamlines Figure 22: Computed and Experimental Pressures


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E: The Pseudo Shock Train Problem A characteristic of supersonic internal flows is the development of a strong shock wave that forms as a result of

the flow interaction with the boundary layer along the wall surface of the device. The shock is bifurcated and a series of repeated shocks is formed along the flow passage. The number of these shock series and their position in the flow passage is dependent on the freestream Mach number, the pressure conditions imposed at both the upstream and downstream locations of the flow passage, the passage geometry, and the wall friction due to viscosity. This phenomena was described in Om et al25 as ‘multiple shocks’, by McCormick18, as a ‘shock system’, and most recently by Carroll and Dutton26, and Matsuo27 as a ‘pseodo-shock train’. Through the interaction region, the flow can be decelerated from a supersonic flow to a subsonic flow. Figure 23 presents an illustration of the shock train phenomenon in an internal duct. The shock train is clearly visible and is usually followed by an adverse pressure gradient region in a long enough duct. The interaction region including both shock train and the subsequent pressure recovery region is referred to as a ‘pseudo shock’27.

Figure 23: Schematic of Shock Train and Pseudo Shock27

Also presented in Figure 23 is the behavior of the static pressure at the wall and at the center line of the constant

area duct. Examination of this figure revels that the static wall pressure rises at a faster rate in the shock train region, i.e. between points 1, j, than in the mixing region, i.e. between points j, 2. A similar trend (pressure rise) is observed in the static pressure distribution along the duct centerline. The oscillations observed in the static pressure distribution along the duct centerline are due to the presence of the successive shock series or shock train. Beyond point j no shock exists and the pressures at the wall surface and at the centerline are essentially the same.

The pressure rise between the points 1 and j is caused by the shock train. If the flow is fully subsonic and uniform at the point j, then the static pressure downstream of this point should decrease due to the frictional effect. However, it should be noted that the shock train is followed by the “mixing region”, where no shocks exist. The pressure increase in this region is due in some extent to the mixing of a highly non-uniform profile created by the shock train. The static pressure reaches its maximum at point 2. The results of this research effort are compared to the analysis conducted in Ref 28 since the input data are taken from the same source. Refer to Figure 24. The results of the IDS study as they relate to the density, velocity and pressure distribution profiles are presented in Figures 25 and 26. Again, it can be seen that the IDS procedure reproduces the expected flow field behavior.


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Figure 24: Velocity vectors imposed on density contours, Ref. 28

Figure 25: IDS generated Velocity vectors imposed on density contours

Figure 26: IDS generated pressure contours for the Pseudo-Shock Train Problem

V. Conclusion In an effort to improve the efficiencies of Computational Fluid Dynamic tools a new numerical computational scheme/framework has been developed and is introduced. The ‘Intergo-Differential Scheme’, IDS is a numerical procedure based on two fundamental types of control volumes, spatial and temporal cells. This new scheme was tested and validated on a chosen set of fluid dynamics problems and the results presented above. These problems are listed as follows: the Hypersonic Leading Edge problem, the incompressibility Lid Driven Cavity problem, the Shock Boundary Layer Interaction Problem, and the Isolator Pseudo-Shock Train problem. The solutions obtained for these problems using the IDS procedure showed very good agreement with the physical expectations


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and experimental data for each problem. A survey of CFD literature indicated that a large percentage of the compressible flow field solvers encountered convergence difficulties when utilized to solve incompressible flow problems29. In an effort to demonstrate the capability of the IDS scheme the incompressible lid driven cavity problem was addressed. The IDS scheme delivered excellent results for this problem without any convergence difficulties. In addition, the IDS scheme was used to solve a pseudo-shock train problem which is considered among one of the most challenging problems encountered in computational fluid dynamics29. Once again, the IDS procedure predicted excellent results. In conclusion, the solutions obtained for compressible external and internal flows, and the incompressible lid driven cavity problem validates the IDS procedure as a robust, efficient, and capable scheme that can solve a variety of complex fluid dynamics problems.

Acknowledgments This work has been partially sponsored by the following agencies; AFRL/WPAB and NASA Glenn Research Center. In addition, special appreciation is extended to Mr. Donnie Saunders of the Air Vehicles Directorate at Wright Patterson Air Force Base and Dr. Isaiah M. Blankson of NASA Glenn Research Center for their encouragement and support of this project.

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[American Institute of Aeronautics and Astronautics 49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition - Orlando, Florida ()] 49th AIAA