Aerothermal Optimization of Internal Cooling PassagesUsing a Discrete Adjoint Method

Ping He, Charles A. Mader, Joaquim R. R. A. Martins, Kevin J. MakiUniversity of Michigan, Ann Arbor, MI, USA 48109

Aerothermal optimization is a powerful technique for the design of internal cooling passages becauseit maximizes heat transfer and simultaneously minimizes pressure loss. Moreover, the optimizationis fully automatic, which reduces the duration of design process compared with a human-superviseddesign approach. Existing optimization studies commonly rely on gradient-free methods, which canonly handle a small number of design variables. However, cooling passage designs use complex geom-etry configurations (e.g., serpentine channels with rib-roughened surfaces) to enhance heat transfer;what is needed is to parameterize the passage using a large number of design variables. To addressthis need, we perform aerothermal optimization using a gradient-based optimization algorithm alongwith the discrete adjoint method to compute derivatives. The benefit of using the adjoint method isthat its computational cost is independent of the number of design variables. In this paper, we fo-cus on optimizing a U-bend duct, which is representative of a simplified, rib-free turbine internalcooling passage. The duct geometry is parameterized using 135 design variables, which gives ussufficient design freedom for geometric modification. We construct a Pareto front for heat transferenhancement and total pressure loss reduction by running multi-objective optimizations. We alsocompare our optimization results with those from the gradient-free methods and demonstrate thatwe achieve better pressure loss reduction and heat transfer enhancement. The above results showthat our gradient-based optimization framework functions as desired and has the potential to be auseful tool for turbine aerothermal designs with full internal cooling configurations.

I. IntroductionTo increase power and efficiency, gas turbine designs adopt a high turbine inlet temperature that is hundreds of

degrees higher than the upper limit turbine materials can withstand [1]. Therefore, turbine cooling systems are critical,as they have a large impact on the safety and durability of gas turbine engines. One common way to reduce theturbine temperature is to inject cooling air into the turbine blades. The cooling air first flows through a serpentinerib-roughened internal cooling passage, and then flows to other cooling systems, e.g., jet impingement cooling orfilm cooling. The serpentine rib-roughened passage creates strong shear that generates flow separation and vortices toenhance turbulence mixing and heat transfer. However, the separated flow also increases the total pressure loss, whichhas an adverse impact on the effectiveness of jet impingement and film cooling. Therefore, turbine internal passagedesign needs to balance heat transfer enhancement and pressure loss reduction, and aerothermal design is required [1].

Owing to the complex geometry in the serpentine cooling passages with full rib configurations, traditional aerother-mal designs have relied heavily on experiments [1]. More recently, however, computational fluid dynamics (CFD) hasbecome a powerful tool to simulate the flow and heat transfer characteristics and provide useful design insights. Tofurther increase the CFD as a design tool, we use an optimization algorithm to automate the design process. A recentstudy of turbomachinery aerodynamics demonstrated that the CFD-based design optimization approach reduced theduration of design process by 85%, while achieving a same quality compared with a sophisticated human-superviseddesign tool [2]. Given this advantage, CFD-based design optimization is becoming increasing popular.

There are two main classes of design optimization methods: gradient-free and gradient-based methods. A numberof researchers have optimized turbine internal cooling designs using the gradient-free methods such as artificial neuralnetwork, genetic algorithms, and design of experiments [3–6]. However, one of the issues with these gradient-freemethods is that their computational cost scales exponentially with the number of design variables [7]. Therefore, theabove studies only parameterized the cooling configuration using a small number of design variables (ranging from 3to 26). Given the complex geometry of a serpentine, rib-roughened cooling passage, it is preferable to parameterizethe passage using a large number of design variables to achieve the maximum possible performance. We addressthis limitation by using a gradient-based optimization method together with efficient gradient computation techniques.Instead of evaluating a large number of CFD samples, the gradient-based method starts from the baseline design andutilizes the gradient information to find the most promising direction in the design variable space for improvement.To efficiently compute the derivatives, we utilize the adjoint method whose computational cost is independent of thenumber of design variables [8, 9].

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2018 Joint Thermophysics and Heat Transfer Conference

June 25-29, 2018, Atlanta, Georgia

10.2514/6.2018-4080

Copyright © 2018 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

AIAA AVIATION Forum

This combination of gradient-based optimization with adjoint derivative computation has been used in aerody-namic designs for aircraft [10–15], cars [16, 17], and turbomachinery [18, 19], as well as multidisciplinary designoptimization [14, 20] such as aerostructural [21–23] and hydrostructural [24] optimization. The adjoint method hasalso been utilized to optimize the heat transfer performance for turbine blades and fin heat exchangers [25, 26]. In thecontext of turbine internal cooling passage design, the adjoint method has not been widely used. We believe this is pri-marily because the flow structure in the serpentine rib-roughened passages is highly complex, as mentioned above. Asa result, the flow converges poorly and the accuracy of adjoint derivatives is degraded. Moreover, when a strong flowseparation is present, the adjoint fails to converge and the optimization aborts. Adjoint optimization under poor flowconvergence conditions is known to be a challenging issue [16], and advanced flow and adjoint stabilization techniqueis needed.

Despite the above challenge, we believe that the adjoint method is the only hope for the aerothermal optimizationof serpentine cooling passages with full rib configurations, where hundreds of design variables are needed. As a firststep in this direction, we apply the adjoint method to perform the shape optimization of an internal cooling passagedesigns. There are two types of adjoint implementations: continuous and discrete [27]. We opt to use the discreteadjoint approach because the adjoint derivative is consistent with the flow solution. Moreover, the discrete adjointimplementation is easier to maintain and extend (for example, when adding new objective or constraint functions andboundary conditions). We conduct aerothermal optimizations for a U-bend duct that is representative of a simplifiedrib-free turbine internal cooling passage. We demonstrate the capability of our adjoint optimization framework byusing 135 design variables to parameterize the U-bend duct. Moreover, we evaluate the benefit of the adjoint methodby comparing our optimization results with those generated by the gradient-free methods.

The reminder of this paper is organized as follows. In Section II, we introduce the optimization framework and itscomponents. The aerothermal optimization results for the U-bend duct are described in Section III. We end with ourconclusions in Section IV.

II. MethodIn a previous study, we implemented an aerodynamic design optimization framework using a discrete adjoint

method with OpenFOAM [17]. In this study, we extend this framework for aerothermal optimization. The centralelement in our design optimization framework is a discrete adjoint solver for computing the total derivative df/dx,where f is the function of interest (objective and constraint functions, e.g., Nusselt number, total pressure loss),and x is the design variable vector that controls the geometric shape via free-form deformation (FFD) control pointmovements [28]. Our framework also depends on the following external libraries and modules: open field operationand manipulation (OpenFOAM) [29, 30], portable, extensible toolkit for scientific computation (PETSc) [31, 32],pyGeo [28], pyWarp [28], and pyOptSparse [33]. In this section, we briefly introduce the overall adjoint optimizationframework, the flow solution technique, and the adjoint derivative computation method. The detailed theoreticalbackground for the other modules was reported in our previous work [17].

1. Discrete Adjoint Optimization FrameworkThe modules and data flow for the optimization framework are shown in Fig. 1, where we use the extended designstructure matrix (XDSM) standard developed by Lambe and Martins [34]. The diagonal entries are the modules inthe optimization process, while the off-diagonal entries are the data. Each module takes data input from the verticaldirection and output data in the horizontal direction. The thick gray lines and the thin black lines denote the data andprocess flows, respectively. The numbers in the entries represent their execution order.

The framework consists of two layers: OpenFOAM and Python. The OpenFOAM layer consists of a flow solver(buoyantBoussinesqSimpleFoam), an adjoint solver (discreateAdjointSolver), and a graph coloring solver (coloring-Solver). The flow solver is based on the OpenFOAM built-in solver buoyantBoussinesqSimpleFoam for steady incom-pressible turbulent flow with heat transfer. The adjoint solver computes the total derivative df/dx based on the flowsolution generated by buoyantBoussinesqSimpleFoam. To accelerate the partial derivative computation in the adjointsolver, we developed a parallel graph coloring solver[17].

The Python layer is a high-level interface that takes the user input, as well as the total derivatives computedby the OpenFOAM layer, and calls multiple external modules for optimization. More specifically, these externalmodules are: “pyGeo” for geometric parameterization, “pyWarp” for volume mesh deformation, and “pyOptSparse”for optimization problem setup. A brief summary of these three modules follows.

pyGeo: a surface geometry parameterization module that implements the FFD approach [28]. This approach embedsthe geometry into a volume such that the geometry is manipulated by moving points at the surface of that volume

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x(0) Graph Coloring

x(∗) 0, 5→1:pyOptSparse

1 : x

5 : c, dc/dx1:

pyGeo2 : xS

2:pyWarp

3 : xV 4 : dxV /dx

5 : f3:

buoyantBoussinesq-SimpleFoam

4 : w

5 : df/dx4:

discreteAdjointSolver

Figure 1. Extended design structure matrix [34] for the discrete adjoint framework for constrained shape optimization problems. x:design variables; x(0): baseline design variables; x(∗): optimized design variables; xS : design surface coordinates; xV : volume meshcoordinates; w: state variables; c: geometric constraints; f : objective and constraint functions.

Figure 2. FFD control points for the U-bend duct. We move only the red points, while the black points remain unchanged during optimiza-tion.

(the FFD points). The FFD volume is a tri-variate B-spline volume such that the gradient of any point inside thevolume can be easily computed. An example of FFD control points is shown in Fig. 2.

pyWarp: a volume mesh deformation module using the analytical inverse distance algorithm [35]. The advantageof this approach is that it is highly flexible and is applicable for both structured and unstructured meshes. Inaddition, compared with the radial basis function based method, this approach does a better job of preservingmesh orthogonality in the boundary layer.

pyOptSparse: an open source, object-oriented Python module, extended from pyOpt [33], for formulating and solv-ing constrained nonlinear optimization problems. pyOptSparse provides a high-level API for defining the designvariables, and the objective and constraints functions. It also provides interfaces for several optimization pack-ages, including some open source packages. In this study, we choose SNOPT [36] as the optimizer. SNOPT

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implements a sequential quadratic programming algorithm to solve the constrained nonlinear optimization prob-lem and uses the quasi-Newton method to solve the quadratic subproblem, where the Hessian of the Lagrangianis approximated by using a Broyden–Fletcher–Goldfarb–Shanno update.

2. Flow SolutionIn this paper, we simulate three-dimensional steady turbulent flow with heat transfer using the OpenFoam built-insolver buoyantBoussinesqSimpleFoam. The flow is governed by the Navier–Stokes equations and the temperatureequation

∇ ·U = 0, (2.1)∇ ·UU +∇p−∇ · (ν + νt)(∇U +∇UT ) = 0, (2.2)

∇ · TU −∇ · (α+ αt)∇T = 0, (2.3)

where p is the pressure, T is the temperature, U=[u, v, w] is the velocity vector representing components in the x, y,and z directions, respectively, ν and νt are the molecular and turbulent viscosity, respectively, and α and αt are themolecular and turbulent thermal diffusivity, respectively. The viscosity and thermal diffusivity are connected throughPr = ν/α = 0.7, and Prt = νt/αt = 0.85. We ignore any body forces and external or internal heat sources. Toconnect the turbulent viscosity to the mean flow variables, we use the k − ω SST turbulence model. We include allthe turbulence variables in our adjoint implementation; we do not assume “frozen turbulence” as is typically done incontinuous adjoint implementations [37].

We use the finite-volume method to discretize the above governing equations. To be more specific, we use the semi-implicit method for pressure-linked equations (SIMPLE) algorithm [38] to solve Eqs. (2.1) to (2.3) in a segregatedmanner. We first solve the momentum and temperature equations based on the old p and φ fields, where φ is thesurface flux. Then p is updated by solving a pressure Poisson equation, followed by an update for φ using the Rhie–Chow interpolation scheme [39]. Based on the new p and φ, we then update U such that it satisfies both mass andmomentum equations.

3. Discrete Adjoint Derivative ComputationAs mentioned above, we need to compute the total derivative df/dx to perform gradient-based shape optimization,where f depends not only on the design variables, but also on the state variables that are determined through thesolution of governing equations. Thus,

f = f(x,w), (3.1)

where w is the vector of state variables. Applying the chain rule for the total derivative, we obtain

df

dx=∂f

∂x+∂f

∂w

dw

dx. (3.2)

A naive computation of dw/dx via finite-differences requires solving the governing equations nx times, where nx isthe number of design variables. This operation is computationally expensive for a large number of design variables.We avoid this issue by using the fact that the derivatives of the residuals with respect to the design variables must bezero for the governing equations to remain feasible with respect to variations in the design variables. Thus, applyingthe chain rule to to the residuals, we can write

dR

dx=∂R

∂x+∂R

∂w

dw

dx= 0, (3.3)

where R is the vector of flow residuals. Substituting Eq. (3.3) into Eq. (3.2) and canceling out the dw/ dx term, weget

df

dx=∂f

∂x− ∂f

∂w

(∂R

∂w

)−1∂R

∂x. (3.4)

Considering the combination of the ∂R/∂w and ∂f/∂w terms in Eq. (3.4), we solve a linear equation

∂R

∂w

T

ψ =∂f

∂w

T

(3.5)

to obtain the adjoint vector ψ. Then, this adjoint vector is substituted into Eq. (3.4) to compute the total derivative

df

dx=∂f

∂x−ψT ∂R

∂x. (3.6)

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Figure 3. Structured mesh for the U-bend duct.

Since the design variable x does not explicitly appear in Eq. (3.5), we need to solve Eq. (3.5) only once for eachfunction of interest, and thus the computational cost is independent of the number of design variables. This is anadvantage for three-dimensional shape design optimization problems, since the number of functions of interest isusually less than 10 but the number of design variables can be a few hundred.

A successful implementation of adjoint-based derivative computation requires an efficient and accurate computa-tion for the partial derivatives (∂R/∂w, ∂R/∂x, ∂f/∂w, and ∂f/∂x) in Eqs. (3.5) and (3.6). In this paper, we usethe finite difference to compute the partial derivatives. To accelerate the finite-difference based partial derivative com-putation, we utilize a heuristic parallel graph coloring algorithm [17]. We partition all the columns of the Jacobiansinto different structurally orthogonal subgroups (colors), such that, in one structurally orthogonal subgroup, no twocolumns have a nonzero entry in a common row. With this treatment, we simultaneously perturb multiple columnsthat have the same colors. The number of residual evaluation is reduced to O(1000), independent of the mesh size andnumber of CPU cores [17].

As mentioned above, in this paper, we extend our adjoint-based aerodynamic optimization framework [17] foraerothermal optimization. Similar to our previous aerostructural adjoint implementation [21], we formulate theaerothermal adjoint system using the multidisciplinary feasible (MDF) approach [14]. To be more specific, we firstsolve an aerothermal problem using the segregated method showed in Section 2. Based on the converged flow, weassemble the coupled adjoint system including all state variables, i.e., w=[U , p, T , k, ω, φ]. We then solve the cou-pled adjoint system using the generalized minimal residual method. Compared with other multidisciplinary designoptimization architectures such as individual discipline feasible (IDF) and simultaneous analysis and design (SAND),the benefit of using the MDF approach is that the multidisciplinary system has a minimal number of control variables(i.e., design variables, objective and constraint functions) for the optimizer. Moreover, it always returns a system thatsatisfies the governing equations and constraints, even the optimization terminates early [14].

III. Results and DiscussionIn this section, we first validate our flow solver by comparing our simulation results with experiments. We then

perform a series of optimizations for different objectives and discuss the optimization results.

1. ValidationAs mentioned in the introduction, we use the U-bend duct as our benchmark case, whose geometry is shown in Fig. 2.Since the duct is symmetric with respect to the z=0 plane, we simulate only half of the geometry. We generate astructured mesh with 0.25 million cells, as shown in Fig. 3. The average y+ is 2.0. Following Coletti et al. [40], we setthe inlet velocity to 8.4 m/s with a turbulence intensity of 5%. The Reynolds number is 4.2×104 based on the hydraulicradius (Dh=0.075 m). The inlet temperature is 293.15 K, while the wall temperature is fixed at 303.15 K with a 10 Ktemperature difference driving the heat transfer. The symmetric boundary condition is imposed at the z=0 plane. Theturbulence model is k−ω SST. As mentioned in the introduction, large regions of separated flow are typically present

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Figure 4. The simulation results qualitatively agree with the experiments [40]. However, the velocity is underestimated at the symmetryplane, and Nu is overestimated at the upper and outer surfaces.

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Table 1. Optimization configuration for the U-bend duct. We use 135 design variables to parameterize the duct geometry.

Function or variable Description Quantityminimize f = w1CPL − w2Nu weighted Nu and CPL

with respect to ∆x displacement of FFD points 135

subject to gsymmy =0 y-direction slope at symmetry plane is zero 18−0.05 m < ∆x < 0.05 m design-variable bounds

in the U-bend duct. Therefore, to ensure a good flow and adjoint convergence during the optimization process, we usethe first-order upwind scheme to discretize the inviscid terms, and central differencing for the viscous terms. From aCFD perspective, the first-order scheme introduces strong numerical dissipation and is not recommended. However,from an optimization standpoint, it is hopeful that the adjoint derivatives based on the first-order scheme qualitativelycapture the correct trend for improvement. Adjoint optimization using the first-order discretization scheme has beenused in previous studies where large flow separation is present [41, 42].

Now we validate our CFD solver by comparing the simulated velocity field with the experimental data of Colettiet al. [40], as shown in Fig. 4. Note that our definition of z/Dh differs from Coletti et al. [40] due to the use of thesymmetry plane. The velocity distribution at the symmetric (z/Dh=0) and upper (z/Dh=0.94) planes qualitativelyagree with the experiments. However, the flow separation and velocity magnitude are underestimated, especially atthe symmetry plane.

Next, we compare the distribution of normalized Nusselt number (Nu/Nu0). Nu is defined as

Nu =QDh

(Tw − TB)k, (1.1)

where Q is the heat flux, k is the thermal conductivity, and Tw and TB are the wall temperature and bulk temperature,respectively. To compute the bulk temperature, we assume a linear increase in mainstream temperature from theinlet to outlet. We then linearly interpolate TB based on the normalized streamwise location. The reference Nusseltnumber Nu0 is computed based on the Dittus–Boelter correlation (Nu0 = 0.023Re0.8Pr0.4). As shown in Fig. 4,the distribution of simulated Nusselt number qualitatively agrees with the experiments. We observe high Nu in thedownstream region of U-bend, which is due to the flow separation and enhanced turbulence mixing. However, thesimulation overestimates the magnitude of Nu at both top and outer walls.

2. Aerothermal OptimizationTo consider both aerodynamics and heat transfer in the U-bend duct design, we define an objective function using aweighted sum of Nu and CPL,

f = w1CPL − w2Nu, (2.1)

where w1 and w2 are weights (w1 + w2 = 1), and CPL is defined as

CPL =pin0 − pout

0

0.5ρU20

. (2.2)

Here, pin0 and pout0 are the total pressure at the inlet and outlet, respectively, ρ is the density, and U0 is the inlet velocity.In Eq. (2.1), we use relative values for Nu and CPL; Nu and CPL are normalized with respect to the baseline designvalues.

We set 63 FFD control points to manipulate the shape of U-bend, as shown in Fig. 2. Only the red FFD pointsare selected as the design variables, while the black FFD points remain unchanged during the optimization process.The FFD points on the upper surface move in the z direction, while the points on the inner and outer surfaces movein the x and y directions. In total, we have 135 design variables. To ensure a smooth geometric transition at thesymmetry plane, we enforce a zero-slope constraint for the first and second layers of FFD points in the z direction.The optimization problem is summarized in Table 1.

To construct a Pareto front for Nu and CPL, we run optimizations using six combinations of weights, as shownin Table 2. From Opt0 to Opt5, we gradually increase the weights for CPL. Opt5 is equivalent to a min-CPL case.For the max-Nu case (Opt0), if we set w1=0, the optimizer tends to generate large flow separation to enhance heattransfer. As a result, the flow convergence is poor and the adjoint fails to converge. Therefore, we give a 2% weight

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Table 2. Weights for multi-objective optimization.

Opt0 Opt1 Opt2 Opt3 Opt4 Opt5w1 0.02 0.05 0.10 0.20 0.50 1.00w2 0.98 0.95 0.90 0.80 0.50 0.00

-100 0 100 200 300 400 500∆CPL, %

-30

-20

-10

0

10

20

30

40

50

∆Nu

, %

Baseline

Opt0

Opt1

Opt2

Opt3

Opt4

Opt5

Opt0: ∆Nu= 40.3%, ∆CPL= 438.5% Opt1: ∆Nu= 27.9%, ∆CPL= 170.3% Opt2: ∆Nu= 15.0%, ∆CPL= 33.8% Opt3: ∆Nu= 3.6%, ∆CPL= -20.6% Opt4: ∆Nu= -14.2%, ∆CPL= -48.1% Opt5: ∆Nu= -23.8%, ∆CPL= -52.1%

Figure 5. Pareto front for Nu and CPL. For the min-CPL case (Opt5), we obtain 52.1% CPL reduction, while for the max-Nu case, weincrease Nu by 40.3%.

for CPL. This allows the optimizer to maximize Nu while maintaining the flow separation at a level such that theoptimization can proceed.

We run all the optimizations on TACC Stampede2 using SKX nodes. The SKX nodes are equipped with IntelXeon Platinum 8160 CPU running at 2.1 GHz. For each optimization, we use 96 CPU cores with 4 SKX nodes. TheOpt0 to Opt5 cases converge in 41, 46, 30, 25, 16, and 24 iterations. On average, each iteration takes 300 s. Theoptimization requires more iterations to converge if a higher weight is given to Nu. This is expected, as the optimizertends to create flow separation to enhance heat transfer. As a result, the accuracy of adjoint derivatives degrades. Thiseventually increases the number of optimization iterations. We observed a similar behavior in our previous study [17].

Figure 5 shows the Pareto front generated from the multi-objective optimization runs. For the Opt5 case, we obtain52.1% CPL reduction. Our adjoint-based CPL reduction is much higher than the value (36%) reported by Verstraeteet al. [43], who used a gradient-free method (combined ANN and kriging). This extra CPL reduction is primarilyattributed to the use of a large number of design variables. Since the turbulence mixing is suppressed in the duct,Nu decreases by 23.8%. When increasing w2 to 0.98 (Opt0), we obtain a 40.3% increase in Nu. Again, we obtainmuch more Nu increase, compared with the value (16.9%) reported by Verstraete et al. [4]. However, this design isnot favorable, since it generates excessive flow separation. As a result, CPL increases by 438.5%. For the Opt3 case,we obtain a 3.6% increase in Nu and a 20.6% reduction in CPL; a favorable design for both aerodynamics and heattransfer.

The velocity distribution is closely correlated with the pressure loss and heat transfer. As the flow accelerates,the friction loss increases. In the meantime, the increased velocity also increases the convective heat transfer. Withthis in mind, we now compare the baseline and optimized shapes for Opt0, Opt3, and Opt5, as shown in Fig. 6. Thecorresponding velocity contour and streamline at the symmetry planes are shown in Fig. 7. For Opt5, the U-bendexpands, and the velocity decreases. Therefore, the friction loss is reduced. On the other hand, for Opt0, the ductshrinks before the U-bend section, forcing the flow to accelerate. This also generates a large separation region in thedownstream of U-bend, as shown in Fig. 7. For Opt3, the U-bend also expands, which reduces the flow separationcompared with the baseline case. However, the velocity magnitude is maintained.

Figures 8 and 9 show the Nu contour on the duct surfaces. The average streamwise Nu is plotted in Fig. 10 forreference. As shown in Fig. 10,Nu increases in the U-bend section and then gradually decreases for all configurations.For Opt0, due to the flow acceleration, Nu/Nu0 increases up to 4.2 in the U-bend section. The increased Nu is most

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Figure 6. Comparison between the baseline and optimized shapes. For the min-CPL case (Opt5), the bend section expands, while for themax-Nu case (Opt0), the duct shrinks before the bend section.

Figure 7. Velocity contour and streamline at the symmetry planes. For the min-CPL case (Opt5), velocity decreases due to the ductexpansion, while for the max-Nu case (Opt0), velocity increases and a large separation is found downstream of the bend section.

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Figure 8. Comparison of Nu between the baseline and optimized shapes. For the min-CPL case (Opt5), Nu decreases due to the reducedvelocity in the duct, while for the max-Nu case (Opt0). Nu is large on the top and outer surfaces due to the increased velocity.

evident on the top and outer walls in the downstream region of U-bend (Figs. 8 and 9). For Opt5, Nu decreasescompared with the baseline case at all streamwise locations. As discussed before, this is because the duct expands andthe velocity reduces. The streamwise Nu distribution for Opt3 is similar to the baseline design, with an average Nuincrement of 3.6%.

IV. ConclusionIn this paper, we develop the capability to perform aerothermal optimization for internal cooling passages using

a gradient-based optimization algorithm along with the discrete adjoint method for computing derivatives. Comparedwith approaches based on gradient-free methods, our approach can handle a much larger number of design variables,and thus gives us more design freedom for geometric modification. We run multi-objective optimizations for a U-bend duct representative of a simplified rib-free turbine internal cooling passage. We consider both heat transferenhancement and pressure loss reduction, and explore different combinations of weights for these two objectivesduring the optimizations, varying from max-Nu to min-CPL. The duct geometry is parameterized using 135 designvariables. We demonstrate our optimization capability by comparing our optimization results with the ones based onthe gradient-free methods. We obtain 52.1% CPL reduction for the min-CPL case, which is higher than the previouslyreported value (36%), where a gradient-free optimization method was used. For the max-Nu case, we achieve a 40.3%increase in Nu. This value is also higher than the previously reported value (16.9%). The above results highlight thatour adjoint optimization framework is efficient and effective. Our optimization framework has the potential to be auseful tool for aerothermal designs of turbine internal cooling passages with full rib configurations.

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Figure 9. Same as Fig. 8 from a different view.

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Streamwise Location

0

1

2

3

4

5

Nu/Nu

0

U Bend Section

BaselineOpt0Opt3Opt5

Figure 10. Average Nu along the streamwise location. For the max-Nu case (Opt0), Nu/Nu0 increases to 4.2 in the duct section.

AcknowledgmentsThe computations were done in the Extreme Science and Engineering Discovery Environment (XSEDE), which is

supported by National Science Foundation Grant No. ACI-1548562.11 of 14

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