NASA/TM—2015-218932

Utilizing Direct Numerical Simulations of 

Transition and Turbulence in Design 

Optimization 

Man M. Rai

Ames Research Center

Moffett Field, CA-94035

December 2015
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NASA/TM-2015-218932

Utilizing Direct Numerical Simulations of 

Transition and Turbulence in Design 

Optimization 

Man M. Rai

Ames Research Center

Moffett Field, California

National Aeronautics and

Space Administration

Ames Research Center

Moffett Field, California 94035-1000

December 2015
2

UTILIZING DIRECT NUMERICAL SIMULATIONS OF TRANSITION 

AND TURBULENCE IN DESIGN OPTIMIZATION 

Man Mohan Rai

Ames Research Center

SUMMARY

Design optimization methods that use the Reynolds-averaged Navier-Stokes equations with the 

associated turbulence and transition models, or other model-based forms of the governing 

equations, may result in aerodynamic designs with actual performance levels that are noticeably 

different from the expected values because of the complexity of modeling turbulence/transition 

accurately in certain flows. Flow phenomena such as wake-blade interaction and trailing edge 

vortex shedding in turbines and compressors (examples of such flows) may require a 

computational approach that is free of transition/turbulence models, such as direct numerical 

simulations (DNS), for the underlying physics to be computed accurately. Here we explore the 

possibility of utilizing DNS data in designing a turbine blade section. The ultimate objective is to 

substantially reduce differences between predicted performance metrics and those obtained in 

reality. The redesign of a typical low-pressure turbine blade section with the goal of reducing 

total pressure loss in the row is provided as an example. The basic ideas presented here are of 

course just as applicable elsewhere in aerodynamic shape optimization as long as the 

computational costs are not excessive.

INTRODUCTION

Computational fluid dynamics (CFD) analyses of propulsion related flows are used in both 

assessing designs and in formal design optimization efforts to compute performance metrics for 

any given geometric configuration. These analyses are currently performed with the Reynoldsaveraged Navier-Stokes (RANS) equations. On occasion large-eddy simulations (LES) are 

performed to more accurately compute the underlying physics. Turbulence models are required 

in both these approaches. These “model-based” computational techniques yield approximations 

to the true flow physics. Direct numerical simulations (DNS) of turbulent/transitional flows 

capture all the essential scales of such flows but require grids that provide accurate resolution of 

the smaller scales (typically about 15 to 25 times as large as the Kolmogorov length scale in 

turbulent flow) while simultaneously encompassing the entire computational region. In highReynolds number flows the smaller scales are significantly smaller than the largest scales and 

the computational region. This disparity in scales necessitates large grids and consequently, 

large computational resources. To date, for the most part, DNS has been applied to relatively 

simple geometries and flows with low to moderate Reynolds numbers.

There are many phenomena in propulsion related flows that may require DNS in order to 

truly capture the underlying physics. Wake-blade interaction and vortex shedding at trailing 

edges of blades in rotating machinery are two examples of flow features whose accurate 

computation may require an approach that does not rely turbulence/transition modeling. These 
3

flow features contribute significantly to the acoustics, unsteady heat transfer, performance 

losses and dynamic loads on the machine and can thus may have a significant effect on 

efficiency, reliability, safety and maintenance costs. Here we illustrate this point using a past 

example involving the Space Shuttle Main Engine (SSME). The system under consideration 

was an axial turbine with liquid oxygen as the working fluid, the Low Pressure Oxidizer TurboPump (LPOTP). Inspection of the first row of vanes in this turbine showed evidence of high 

cycle fatigue (HCF) at the trailing edge near the end-walls. CFD analysis of the known sources 

of HCF indicated vortex shedding as the most probable cause1. It was found that the shedding 

frequency range overlapped the vane trailing edge flapping mode natural frequency. At the 

time, vanes in the first row of the LPOTP were replaced at carefully monitored time intervals 

thus ensuring the safety of the Shuttle flights1. A first attempt at redesigning the vane is reported 

in Ref. 1. The objectives pursued were increased vane strength, decreased shedding amplitude, 

decoupling of the shedding and vane natural frequencies, minimal impact on downstream rows 

and performance robustness to manufacturing tolerances. The design assessment1 indicated 

that these objectives were achieved in substantial measure. 

Given the complexity of the underlying physics of the near/intermediate wake, four different 

computer codes and turbulence models were used1 to compute the flow around the baseline 

and optimized airfoils to assess the reduction in fluctuating amplitude of the surface pressure on 

the trailing edge of the vane and the impact on the dynamic loads on the downstream rotor. All 

of the codes solve the RANS equations in conjunction with various turbulence models to provide 

time-accurate simulations of the flow through the LPOTP turbine. These turbulence models 

were validated and used effectively for several classes of flows over a period of years prior to 

the assessment reported in Ref. 1. However, the flow complexities of the very near wake (less 

than three trailing edge diameters downstream of the blunt trailing edge, see for example Refs. 

2 - 4) and limitations of the turbulence models in computing this region of the wake must be kept 

in mind in evaluating the results presented in Ref. 1. A remarkable finding in Ref. 1 is that all of 

the codes and corresponding turbulence models indicate essentially the same qualitative trends 

in flow quality and significant performance improvements for the redesigned airfoil. However 

these RANS simulations were only used to provide a preliminary assessment of the wake 

characteristics. Numerical simulations that accurately capture the complex physics of wakes 

and detailed experiments are required to fully understand the wakes of the baseline and 

optimized airfoils and their interaction with the downstream rows. Among the computational 

approaches that are available for this exercise, direct numerical simulations are particularly 

suitable because they do not require any modeling of turbulence and transition. The Reynolds 

numbers encountered in rotating machinery are typically moderate (less than 2.0x106 based on 

blade chord) and much smaller compared to those found in external flows. Thus simulating the 

flow in single-stage turbomachinery via DNS on modern supercomputers is possible, albeit 

computationally expensive.

An excellent overview of transitional and turbulent flow in gas turbine engines can be found 

in Ref. 5. A description of the different modes of transition that are encountered – natural, 

bypass, separated flow, periodic unsteady and reverse transition is provided there. Of particular 

interest is that transition can be multi-modal, that is, multiple modes of transition can co-exist at 

the same instant in time but in different regions of the airfoil surface. The critical role played by 

turbulent spots in the transition process and the role of the intermittency function in computing 

transitional flow properties and its dependency on spot production rate are discussed at length. 

The effects of freestream turbulence and pressure gradient on spot production rate are also 

presented here. In Ref. 6 data from extensive testing of multistage axial turbines and 

compressors are used to obtain an understanding of transition processes in embedded stages. 
4

The effects of Reynolds number, loading, turbulence intensity and frequency of wakes and, 

clocking of airfoils in alternate rows on the transition process are discussed. Substantial 

emphasis is placed on defining two relatively distinct periods; one in which transition occurs 

because of a strong interaction between the wake and the airfoil boundary layer (wake-induced 

path, predominantly bypass transition) and the other when this is not the case (path between 

wakes, possible multimodal transition). The paper also discusses the becalmed region behind 

turbulent spots and strips and the effect of this calming on flow transition and separation. Much 

of the earlier work in boundary layer transition in turbomachines is reviewed in Refs. 5 & 6.

In a single stage low-speed turbine, interaction effects on the upstream stator are primarily 

due to the potential flow effect (caused by the relative motion between stator and rotor airfoils). 

On the other hand the downstream rotor is subjected to wake-blade interaction, interaction 

between the secondary flows of the stator with the rotor, and as in the case of the stator, the 

potential flow effect. A true stage simulation that includes both rotors and stators in a single 

coupled computation is required to better understand the complexities of turbine and 

compressor flows. Aspects such as the interaction of the secondary flows of upstream rows on 

downstream airfoils and multi-stage turbomachinery flow physics need a coupled rotor-stator 

approach. A number of RANS computations of rotor-stator interaction have been reported in the 

literature. Some of the earlier work is discussed in Refs. 7 - 13 and the references cited therein. 

Since then there have been a large number of RANS simulations reported and such simulations 

are used in industry for design as well as assessment. In fact even three-dimensional multistage 

simulations with the actual number of airfoils in each row are occasionally performed as 

required. As indicated earlier, although these simulations are configured to include most if not 

all the geometrical complexity found in turbomachines, and additionally even include cooling 

flows and aero-elastic effects, they are limited by the adequacy of the transition and turbulence 

models available today.

MOTIVATION AND OBJECTIVE

Here we provide additional motivation for both using DNS in obtaining the data for 

assessment of existing or new designs, and, in formal design optimization. The first example is 

the SSME LPOTP discussed earlier. The data used in this section is from the published 

literature. 

SSME LPOTP First Vane Redesign

The preliminary flow assessments for the baseline and optimized airfoils of Ref. 1 

(optimized airfoil is referred to as O5) are presented in this section. The computed data 

presented here are from Ref. 14. Flow computations were performed using the Baldwin-Lomax 

turbulence model (one of the models used in Ref. 1). The flow is assumed to be turbulent on 

both the pressure and suction sides of the airfoil because of the high disturbance levels in the 

operating environment. 

The first redesign of the baseline airfoil (baseline is from the LPOTP of the SSME) is 

discussed in Ref. 1. The detailed design requirements are also provided there and are outlined 

below:
5

! Increase the thickness of the airfoil, particularly in the trailing edge region, to both 

strengthen the airfoil and increase its natural frequency corresponding to the trailing 

edge flapping mode

! Reduce trailing edge vortex shedding amplitude 

! Decrease trailing edge vortex shedding frequency to obtain greater separation of 

frequencies (shedding and natural flap mode frequencies)

! Maintain throat area

! Maintain exit flow angle 

! Design a trailing edge that eases the manufacturing process (facilitate metal flow in 

casting)

! Reduce pressure fluctuations on downstream airfoil rows

! Desensitize shedding amplitude to manufacturing tolerances and normal wear and 

tear

The last requirement was added because the preferred manufacturing technique is casting, with 

a large manufacturing tolerance of ±0.006 inches (large in relation to the dimensions of the 

trailing edge).

The baseline airfoil and the optimized airfoil (O5) from Ref. 1 are shown in Fig. 1. Clearly 

the new airfoil is thicker and stronger. A stress analysis1 of the baseline and O5 airfoils showed 

that all relevant measures of stress were lower for O5. The corresponding increase in safety 

factors for O5 ranged from 3.5 to 6.3. It was concluded in Ref. 1 that the part fitted with the O5 

airfoil possessed an essentially infinite life in operation. 

O5 airfoil

Baseline airfoil

Figure 1. A comparison of the baseline and O5 airfoil shapes, Ref. 14.

Figure 2 shows the computed time-averaged pressure distributions on the baseline

and O5 airfoils14. Clearly there is a redistribution of the load. The loading on O5 is more 

uniform. In particular, the loading near the leading edge is higher for O5 than the 

baseline. Additionally in the last quarter chord, where both the airfoils are leaner, O5 

shows a smaller load than the baseline. 
6

0.00 0.25 0.50 0.75 1.00

1.0

p / p

8

t

x/c

Baseline airfoil

O5 airfoil

Figure 2. Distribution of the time-averaged surface pressure, Ref. 14.

Figure 3 shows the computed surface pressure amplitudes14 for the two airfoils 

(caused by trailing edge vortex shedding). The amplitude distribution obtained with O5 is 

lower than that obtained with the baseline airfoil on the entire airfoil surface. In particular, 

O5 shows a reduction of about 75% in peak amplitude (this occurs on the pressure side 

of the trailing edge). 

0.0 0.5 1.0 0.5 0.0

0.00

0.02

0.04

0.06

0.08

0.10

suction side pressure side x/c

Baseline airfoil

O5 airfoil

Trailing edge

(p - p ) / p8max min

Figure 3. Distribution of the unsteady surface pressure amplitude, baseline and 

O5 airfoils, Ref. 14.

Figure 4 shows the computed time variation of the surface pressure14 on O5 and the 

baseline airfoil at the point of maximum fluctuating pressure amplitude. This point occurs 

on the pressure side of the trailing edge for both airfoils. The decrease in amplitude 
7

obtained with O5 is clearly visible here. It can also be observed that O5 yields a lower 

shedding frequency. Figure 5 shows results obtained from a spectral analysis of the 

waveforms14 in Fig. 4. Again, the reduction in amplitude obtained with O5 is clearly 

visible. Additionally, the baseline airfoil sheds at a frequency of 48.6 kilohertz and O5 

sheds at 37.8 kilohertz. Thus O5 results in a reduction of approximately 22% in shedding 

frequency. Based on extensive structural dynamics analyses reported in Ref. 1 it was 

concluded there that this reduction in shedding frequency coupled with an increase in 

the flapping mode natural frequency (for O5) results in a detuning of the two frequencies.

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0 1000 2000 3000 4000

n (number of time steps)

(p - p ) / p

mean

8

Pressure history for the baseline airfoil

Pressure history for the O5 airfoil

Figure 4. Time-variation of pressure at the point of maximum amplitude, baseline and O5 

airfoils, Ref. 14.

Figure 5. Spectral analysis of the pressure variation at the point of maximum 

amplitude, Ref. 14.

.
8

Unsteady rotor-stator computations including the first row of stators (baseline or O5 

airfoils), the downstream rotor row and, the second stator row (downstream of the rotor 

row) are also presented in Ref. 1. These computations show that replacing the baseline 

airfoils in the first row with the O5 airfoils results in a modest improvement or, no 

change, in the flow downstream. The results presented in Refs. 1 & 14 demonstrate that 

the objectives of the optimization effort were achieved to a substantial degree.

However, despite the success indicated in the assessment exercise reported in 

Refs. 1 & 14, the uncertainty in computing the near/intermediate turbulent wakes with 

turbulence models remains. How accurately do the models capture the physics of 

shedding? Are the computed amplitudes and frequencies accurate? Can the computed 

unsteady pressure values on the trailing edge be trusted? Is the computed intermediate 

wake represented well enough to result in accurate predictions of unsteady loads on the 

downstream rotor airfoils? Can the turbulence models be trusted to provide accurate 

trends of unsteady trailing edge loading on the vane for the perturbations in geometry 

that are expected (because of the preferred manufacturing technique)? These questions 

can only be answered by detailed experiments (expensive) or a first-principles approach 

such as DNS. 

Direct numerical simulations of the flow in the first stage of a high-pressure turbine 

and the flow through a single-stage low-pressure turbine are presented in Refs. 15 - 17. 

The computed flow features in these investigations exemplify the complexity of the flow 

in turbine stages and provide additional motivation for including direct numerical 

simulations in the optimization process. Here we explore the possibility of utilizing DNS 

in the optimization of turbomachinery airfoil sections.

APPLICATION OF DNS TO DESIGN OPTIMIZATION

A direct numerical simulation is a first-principles approach to computing flow 

transition and turbulence (model-free). While such simulations are not practical as yet at 

high Reynolds numbers there are problems of engineering interest, where the Reynolds 

numbers are modest, that can be solved via DNS. DNS datasets can be used in many 

ways, the principal ones being a) providing designers with the understanding of the flow 

necessary for developing advanced designs and flow control mechanisms, b) in the 

assessment of new designs, c) providing understanding and data necessary for 

improved turbulence and transition models and d) providing the data for constructing 

response surfaces or computing optimal search directions in design optimization. The 

last of these items is probably the most computationally intensive application of DNS. 

Some in the design/analysis community are of the opinion that given current computing 

capability, using DNS to generate the aerodynamic data within a formal design 

optimization process may be impractical. Here we demonstrate, in the case of 

turbomachinery blading, that there are important design improvements that can be 

obtained with DNS in the design optimization loop (the related computing costs are 

discussed in the following section summarizing the effort).

The case in point is the rotor airfoil in the low-pressure turbine (LPT) stage 

discussed at length in Ref. 17. The performance metric used in the present study is the 

total pressure loss defined as in Ref. 18 (the difference between the total pressure at the 

inlet and that at a point 50% axial chord downstream of the trailing edge, divided by the 
9

inlet value). While the loss is defined as a percentage in Ref. 18 it is not in the present 

study to avoid any confusion with percentage decrease in loss obtained via optimization 

or percentage differences in loss obtained on fine and baseline grids. The total pressure 

values used in this definition are obtained via integration in the tangential direction. The 

Reynolds number based on axial chord and exit flow conditions for the present 

computation is in the range of 75,000 - 100,000. The inlet Mach number is 0.42 and the

average flow-turning angle obtained is 106.7o. All of the computations were performed 

without freestream turbulence.

Figure 6 shows the computational domain used in the DNS; it is divided into four 

zones to facilitate grid generation and provide adequate resolution where it is required. 

The inner “O” grid contains the airfoil. The outer “O” grid interfaces with inner “O” grid 

and the grids in the inlet and exit regions. The computation is three-dimensional; copies 

of the two-dimensional zones shown in Fig. 6 are uniformly spaced in the spanwise (z) 

direction. The grid sizes used in the fine grid computations for assessing the 

performance of the baseline and optimized airfoils are 2181x91x257 (inner O grid), 

999x81x257 (outer O grid), 41x41x257 (inlet grid) and 181x361x257 (exit grid). The total 

number of grid points utilized for assessment is about 89 million. The extent of the 

computational domain in the spanwise direction is 0.3C (C is the axial chord) and the 

wall-normal grid spacing at the airfoil surface is 0.0002C.

Fig. 6. Airfoil geometry and computational zones used in the direct numerical 

simulations.

The high-order accurate upwind-biased method developed in Ref. 19 is used here 

to compute the flow over the airfoils. The inviscid terms are computed using fifth-order 

accurate upwind-biased finite differences. The viscous terms are computed with fourthorder accurate central differences. The method is iterative-implicit in nature, multiple 

iterations are used at each time step to solve the nonlinear finite-difference equations 
10

arising from a fully implicit formulation; the method is second-order accurate in time. 

The boundaries that contain the four computational zones can be broadly classified as 

natural and zonal boundaries. Both the natural and zonal boundary conditions used in 

this study are discussed in Ref. 19. Periodic boundary conditions are imposed on the 

boundaries in the z direction (homogeneity in z). No-slip/adiabatic wall conditions are 

used on the airfoil surface.

The objective here is to reduce total pressure loss by modifying the shape of the 

suction surface. The pressure surface is kept unchanged in this first redesign effort. 

Airfoil parameterization is achieved as in Ref. 20. Total pressure loss was computed via 

DNS for 7 perturbations of these parameters. A model (response surface) was 

constructed to represent the variation of loss with varying values of the shape 

parameters. Details regarding the method of perturbing the parameters and constructing 

the model are provided in Ref. 21. The search for the optimal airfoil was performed using 

the model. To reduce computational costs the direct numerical simulations for the 

perturbed airfoil shapes were performed on a smaller grid than those used for the 

assessment of the baseline and optimal airfoils (the smaller grid is called the standard 

grid henceforth). The grid sizes used in these computations for the perturbed geometries 

are 1401x61x129 (inner O grid), 601x31x129 (outer O grid), 41x41x129 (inlet grid) and 

101x181x129 (exit grid). Thus the simulations for the perturbed airfoil shapes used only 

16 million grid points each. Standard grid simulations (same grid size as above) were 

also obtained for the baseline and optimal airfoils and compared with corresponding fine 

grid simulations in the assessment exercise.

Figure 7a shows the pressure distributions obtained on the standard and fine grids 

for the baseline airfoil. They are in close proximity even though the fine grid has about 

5.5 times as many grid points (with grid refinement in all directions). The total pressure 

loss obtained on the fine and standard grids for the baseline airfoil differs by only about 

6.5%. This comparison of losses and pressure distributions provides us with the 

necessary confidence to utilize the computed standard grid data in constructing the 

model, that is, the data are capable of providing the trends in losses with changing airfoil 

shape (via the model) that are required by the optimizer. A second check wherein the 

total pressure loss for the baseline airfoil (computed on the standard grid) was compared 

with the predicted value from the model resulted in a difference of only 3.8% (the 

baseline airfoil data was not used in obtaining the model). This provides confidence in 

the ability of the model to accurately predict the losses for various airfoil shapes in the 

region in design space where the perturbed airfoils reside (assuming the input data is 

sufficiently accurate), that is, an “inner trust region”. The actual trust region may be 

substantially larger in extent. If the optimal airfoil resides outside the inner trust region in 

design space, the associated flow needs to be computed via DNS to ascertain the 

predicted improvement in performance; here as with the baseline airfoil the 

computations need to be performed on both the standard and fine grids to demonstrate 

the adequacy of the computational grids. This final test was performed in the present 

study. 

Figure 7b shows contours of the time-averaged streamwise velocity (u) and the 

regions of separation on the suction and pressures sides (arrows) obtained for the 

baseline airfoil on the fine grid. These regions are potential areas of flow quality 

improvement for the optimization process.
11

Fig. 7a. Comparison of surface pressure distributions obtained for the baseline 

airfoil on the standard and fine grids.

Fig. 7b. Contours of time-averaged streamwise velocity obtained for the baseline 

airfoil on the fine grid.

As mentioned earlier only the suction side of the airfoil was perturbed and the data 

from DNSs of the flow associated with the perturbed airfoil sections (total pressure at 

50% chord downstream of the trailing edge) were used to construct the model. The 

pressure surface was left unaltered. The model was used to obtain an airfoil with 
12

improved flow characteristics via a gradient search algorithm (for convenience this first 

improved airfoil will be referred to as optimal in the rest of this article). Additional DNSs 

will be required to redesign the pressure surface and also to obtain further refinements 

to the suction surface of the airfoil obtained with this first model. The flow associated 

with the airfoil obtained from the optimization process was computed on both the fine 

and standard grids. The loss coefficients obtained on these two grids differed by only 

1.0%. The closer correspondence than that obtained for the baseline airfoil may be 

because the flow in the case of the optimal airfoil is relatively benign compared to that of 

the baseline airfoil (this feature will be evident in the figures to follow). The computation 

performed on the fine grid for the optimized airfoil showed a decrease in total pressure 

loss of about 43% compared to that obtained on the fine grid for the baseline airfoil. This 

is a significant improvement in performance. It should be noted however that the present 

optimization has been carried out at a single specified freestream turbulence level 

(none), Reynolds number, inlet angle and Mach number. Additional comments regarding 

airfoil optimization for multiple operating points are provided later in this section.

Figure 8 shows contours of the time-averaged streamwise velocity (u, negative 

values only) obtained for the baseline airfoil (Fig. 8a) and the optimized airfoil (Fig. 8b) 

with the fine grid. The minimum and maximum contours levels and the number of 

contour levels plotted are the same in these two figures. 

Fig. 8. Contours of time-averaged negative streamwise velocity in the trailing 

edge region for a) the baseline airfoil and b) the optimized airfoil.
13

Firstly, the flow is not separated on the suction side of the optimized airfoil very near the 

trailing edge. The separation bubble is contained on the suction side. This is not the 

case for the baseline airfoil; the bubble extends beyond the trailing edge and also the 

peak value of negative u is higher. Secondly, the region of separation above the suction 

surface is thicker in the case of the baseline airfoil. Clearly optimization has resulted in 

improved flow quality. The optimized airfoil is more slender in the trailing edge region; 

the radius of the circular trailing edge is the same in both cases.

Figures 9a and 9b show time-averaged contours of fluctuating kinetic energy for the 

baseline and optimized airfoils, respectively. As before, the minimum and maximum

contour levels and the number of contours are the same in these two figures. The 

location of the detached shear layer is shown with an arrow in Fig. 9a. Intensity levels 

increase with increasing x along this shear layer with peak intensities occurring in the 

near wake region. The optimized airfoil shows peak intensity just above the suction 

surface and upstream of the trailing region, indicating end-stage transition or early 

turbulence in the region where the flow reattaches to the suction surface. The peak 

value itself is lower than that obtained for the baseline airfoil. A comparison of the 

regions covered by contour lines (wake spread) in Figs. 9a & 9b shows that the wake is 

thinner for the optimized airfoil. Total pressure profiles in the wake also show the same 

effect.

Fig. 9. Contours of time-averaged fluctuating kinetic energy in the trailing edge 

region for a) the baseline airfoil and b) the optimized airfoil.
14

It is likely that additional optimization steps that involve perturbing the pressure 

surface or minor tweaking of the suction surface will yield even better flow quality and 

lower loss in total pressure. Such an effort will require additional flow simulations and 

associated response surfaces. However, the emphasis in this exploratory effort is a 

simple demonstration that it is practical in some cases (with the computing capability 

currently available) to utilize direct numerical simulations of transitional/turbulent flows at 

low/moderate Reynolds numbers within an optimization loop to improve aerodynamic 

design. Efforts to obtain improved performance over a range of Reynolds numbers and 

different levels of freestream turbulence may result in a multi-objective optimization 

problem and thus require a suitable optimization method22. Designing an airfoil section 

that is relatively insensitive to operating conditions (for example variations in inlet flow 

angle) may require the use of concepts such as robust design optimization22. Satisfying 

these additional constraints may require many more direct numerical simulations than 

used here to obtain the optimal airfoil shape.

CONCLUDING REMARKS

The complexity of turbomachinery flows sometimes precludes the use of RANS 

codes with their associated transition/turbulence models in achieving specific design

objectives and assessing the performance of new designs. Here we explore the use of 

direct numerical simulations in redesigning a turbine blade section for use in 

low/moderate Reynolds number flows. The redesign process required seven standard 

grid simulations (16 million grid points) for constructing a model and two fine grid (89 

million grid points) and two standard grid simulations for assessment. The effort yielded 

an LPT blade section that resulted in improved flow characteristics and a reduction of 

about 43% in total pressure loss (compared to that obtained for the baseline airfoil). Only 

the suction surface of the blade was modified in this exploratory effort. It may be 

possible to further improve flow quality and performance with modifications to the 

pressure surface of the airfoil and additional minor adjustments to the suction surface. 

While the example provided is that of the redesign of a low-pressure turbine blade 

section with the objective of reducing total pressure loss in a single row, the basic ideas 

presented here are of course just as applicable elsewhere in aerodynamic shape 

optimization as long as the computational cost associated with direct numerical 

simulations is not excessive. The optimization presented here was performed on 

different computers using different processors. A rough estimate of the total computing 

time used is 90,000 singe core hours (processors operating at 2.66GHz). The 

experience gained in this exploratory LPT blade section optimization via DNS suggests 

that computational costs for the optimization can be significantly reduced in efforts to 

obtain additional performance improvements.

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Propulsion Conference, Tucson, Arizona, July 10-13, 2005.
15

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Mechanics, Volume 724, 2013, pp. 704-733.

3. Rai, M. M., “Flow Phenomena in the very near wake of a flat plate with a circular 

trailing edge,” Journal of Fluid Mechanics, Volume 756, 2014, pp. 510-543.

4. Rai, M. M., “Detached shear-layer instability and entrainment in the wake of a flat 

plate with turbulent separating boundary layers,” Journal of Fluid Mechanics, Volume 

774, 2015, pp. 5-36.

5. Mayle, R. E., “The Role of Laminar-Turbulent Transition in Gas Turbine Engines,” 

Paper No. 91-GT-282, International Gas Turbine and Aeroengine Congress , Orlando, 

Florida, June 3-6, 1991.

6. Halstead, D. E., Wisler, D. C., Okiishi, T. H., Walker, G. J., Hodson, H. P., Shin, H. 

W., “Boundary Layer Development in Axial Compressors and Turbines, Part 1: 

Composite Picture,” Paper No. 95-GT-461, International Gas Turbine and Aeroengine 

Congress , Houston, Texas, June 5-8, 1995.

7. Rai, M. M., “Navier-Stokes Simulations of Rotor-Stator Interaction Using Patched and 

Overlaid Grids,” Journal of Propulsion and Power, Vol. 3, Sep. – Oct. 1987, pp. 387-396.

8. Rai, M. M., “Three-Dimensional Navier-Stokes Simulations of Turbine Rotor-Stator 

Interaction; Part 1 – Methodology,” AIAA Journal of Propulsion and Power, Vol. 5, No. 3, 

1989, pp. 312-319.

9. Rai, M. M., “Three-Dimensional Navier-Stokes Simulations of Turbine Rotor-Stator 

Interaction; Part 2 – Results,” AIAA Journal of Propulsion and Power, Vol. 5, No. 3, 

1989, pp. 307-311.

10. Rai, M. M., and Madavan, N. K., “Multi-Airfoil Navier-Stokes Simulations of Turbine 

Rotor-Stator Interaction,” Journal of Turbomachinery, Vol. 112, July 1990, pp. 377-384.

11. Gundy-Burlet, K. L., Rai, M. M., Stauter, R. C., and Dring, R. P., "Temporally and 

Spatially Resolved Flow in a Two-Stage Axial Compressor. Part II: Computational 

Assessment." Journal of Turbomachinery, Vol. 113, No. 2, April 1991, pp. 227-232.

12. Madavan, N. K., Rai, M. M., and Gavali, S., “Multipassage Three-Dimensional 

Navier-Stokes Simulation of Turbine Rotor-Stator Interaction,” AIAA Journal of 

Propulsion and Power, Vol. 9, No. 3, 1993, pp. 389-396.

13. Rangwalla, A. A., and Rai, M. M., "A Numerical Analysis of Tonal Acoustics in RotorStator Interactions," Journal of Fluids and Structures, Vol. 7, 1993, pp. 611-637.

14. Rai, M. M., “Designing Compact and Robust Rocket Engine Components for 

Sustainable Space Exploration,” AIAA Paper No. 2006-7098, 11th AIAA/ISSMO 

Multidisciplinary Analysis and Optimization Conference, Portsmouth, Virginia, 

September 6-8, 2006.
16

15. Rai, M. M., “A Direct Numerical Simulation of Transition and Turbulence in a Turbine 

Stage,” AIAA Paper No. 2009-584, 47th AIAA Aerospace Sciences Meeting, January 5-8, 

2009, Orlando, Florida.

16. Rai, M. M., “A Direct Numerical Simulation of Turbine Rotor-Stator Interaction,” AIAA 

Paper No. 2009-3685, 39th AIAA Fluid Dynamics Conference, June 22-25, 2009, San 

Antonio, Texas.

17. Rai, M. M., “A Direct Numerical Simulation of Flow Through a Low Pressure Turbine 

Stage,” AIAA Paper No. 2011-3092, 41st AIAA Fluid Dynamics Conference, June 27-30, 

Honolulu, Hawaii.

18. Medic, G., and Sharma, O., “Large-Eddy Simulation of Flow in a Low-Pressure 

turbine Cascade,” Paper No. GT2012-68878, Proceeedings of the ASME Turbo Expo 

2012: Power for Land, Sea and Air GT2012, June 11-15, 2012, Copenhagen, Denmark.

19. Rai, M. M., “Direct Numerical Simulation of Transitional and Turbulent Flow on a 

Turbine Airfoil,” AIAA Journal of Propulsion & Power, Vol. 26, No. 3, 2010, pp. 587-600.

20. Rai, M. M., and Madavan, N. K., “Aerodynamic Design Using Neural Networks,” 

AIAA Journal, Vol. 38, No. 1, January 2000, pp. 173-182.

21. Rai, M. M., “A Rapid Aerodynamic Design Procedure Based on Artificial Neural 

Networks,” AIAA Paper No. 2001-0315, AIAA 39th Aerospace Sciences Meeting, 

January 8 – 11, 2001, Reno, Nevada.

22. Rai, M. M., “Robust Optimal Aerodynamic Design Using Evolutionary Methods and 

Neural Networks,” AIAA Paper No. 2004-778, AIAA 42nd Aerospace Sciences Meeting, 

January 5 – 8, 2004, Reno, Nevada.