In this study, thermal stresses in a double-wall cooling system are analyzed. We consider an infinite flat double-wall geometry and assume that it can be represented by an axisymmetric unit cell, wherein the thermal loadings and deformation at the boundaries are determined by periodicity conditions. A thermal model is initially developed to obtain the thermal fields using a combination of empirical correlations and computational fluid dynamics (CFD) analysis. The thermal fields are then analyzed such that both the first-order and higher order approximations are determined. A theoretical solution is derived assuming that the temperature gradient takes place only across the outer skin using the first-order approximations. The solution yields an equibiaxial stress state in the skins, which are driven by the thermal curvature of the outer skin. To investigate other geometrical features and higher order approximations, a finite element model is used to solve Fourier’s law of heat conduction and mechanical equilibrium equations. The numerical and theoretical results are found to be in excellent agreement. We determine that the increase of distance between pedestals reduces the stresses. Furthermore, the stress concentration factor at the fillet increases with the increase of both the radius and pedestal diameter and the decrease of the skin thickness. Increasing the number of film holes limits stresses to the external surface of the outer skin. The increase of the Reynolds number in the impingement hole increases the Biot number in the outer skin, which increases the stresses. The higher order approximations of the heat transfer coefficients play a minor role.
Research into quasi-transpiration, double-wall type cooling systems has been ongoing since the 1970s. Their lack of widespread implementation has, in-part, resulted from lack of manufacturing capability and cost, particularly during earlier research into the technology. However, what has proven the most significant challenge in their implementation is the thermomechanical stresses that result in the solid under thermal load due to the large temperature gradients between the two walls .
In this paper, theoretical and numerical analyses of thermal stresses in a double-wall cooling system are presented. The objective is to understand the nature of different stresses due to the large thermal gradient between hot and cold surfaces and to investigate the role of the different geometrical features and heat transfer fields. We consider the flat-plate double-wall geometry studied by Murray et al. . The geometry is assumed to be built from a basic repeating block, i.e., unit cell, such that all cells are subjected to similar thermal loading, i.e., the temperature and heat transfer coefficients fields. In this study, the unit cell is idealized assuming axisymmetric conditions. Consequently, the distributions of the temperature and heat transfer coefficients are taken to be axisymmetric. The mechanical boundary conditions are defined by the periodicity of the unit cell. To obtain the thermal fields, we developed a thermal model using empirical correlations from the literature and the results obtained by Murray et al. . The thermal fields are then analyzed- and first-order and higher order approximations are obtained. The stress analysis of the thermomechanical boundary value problem is then performed in two steps: (i) theoretical solutions are initially derived for simplified thermal fields to understand the nature of the stresses and (ii) numerical analysis using finite element (FE) models of the unit cell to study the geometrical features and higher order approximations of the thermal fields. The problem statement is presented in Sec. 2. The thermal analysis of the double-wall cooling system using empirical correlations and computational fluid dynamics (CFD) analysis are described in Sec. 3. The details of the stress analysis are then formulated in Sec. 4. The stress analysis results for the different geometrical features and heat transfer fields are presented and discussed in Sec. 5.
2 The Problem Statement
In this work, we focus on studying thermal stresses in a double-wall cooling system due to the large thermal gradient between hot and cold surfaces. We consider the flat-plate double-wall geometry studied by Murray et al. . The geometry is built from a basic repeating block, i.e., unit cell, which contains outer and inner skins, impingement holes, film holes, and pedestals, see Fig. 2. The pedestals are of a circular pin-fin geometry. The height and diameter of the pedestals are denoted by H and Dpd, respectively; the spacing between two adjacent pedestals is denoted by L; the impingement hole diameter is denoted by Di; the film hole diameter is denoted by Df; and the outer and inner skin thicknesses are taken to be equal and denoted by t. Furthermore, a pedestal makes two fillets with the outer and inner skins that are assumed to be of equal radius R. The values of these dimensions can be found in Sec. 4.4. The thermal analysis of such a system can be performed using different methods, e.g., full CFD analysis of the cooling air and the mainstream [3,4], a decoupled conjugate approach (using CFD for the internal cooling and empirical correlation to model the mainstream) , or using suitable empirical correlations [5,6]. The outcome of the thermal analysis, using these methods, is the temperature and heat transfer coefficients on the external surface of the outer skin and the internal walls for a given Reynolds number at the impingement holes and mainstream temperature and velocity. Hence, using the knowledge of the temperature and heat transfer coefficients and suitable mechanical boundary conditions, a thermomechanical boundary value problem can be set up, wherein Fourier’s law of heat conduction and mechanical equilibrium equations can be solved to obtain the temperature distribution and stress field in the material.
The main goal of this study is to understand the nature of the stress field and the role of the different geometrical features and internal and external heat transfer factors. We adopt an approach in which the geometry and heat transfer fields are idealized to obtain a theoretical solution of the stress field. More specifically, an axisymetric idealization of the cooling system and suitable thermal and mechanical boundary conditions are assumed, and the details are presented in Sec. 4. Thereafter, appropriate computational models are developed to investigate the complex features of the problem. The basis of this procedure is presented in the following Secs. 2.1–2.3, and the details are provided in Secs. 3 and 4.
2.1 The Geometrical Idealization.
The double-wall cooling system is composed of an infinite array of unit cells. An idealized geometry of this system is shown in Fig. 3(a) where the impingement and film holes are omitted. The unit cell is taken to be the simpler geometry in Fig. 3(b) such that circular sections of the outer and inner skins are connected by a central pedestal. In this analysis, the thermal fields are assumed to be independent of the θ-direction, and the deformation at the boundary of the unit cell is determined by periodicity conditions. Therefore, axisymmetric conditions are assumed to prevail, and henceforth, an axisymmetric representation of the unit cell will be considered.
2.2 The Temperature and Heat Transfer Coefficient Fields.
2.3 The Thermomechanical Boundary Value Problem.
3 The Thermal Model
Literature on the cooling of gas turbines is rife with empirical correlations to predict the Nusselt number for various cooling systems, for example, impingement cooling for various geometries at differing Reynolds numbers and wall spacings or pin-fin/pedestal cooling with different geometrical features and Reynolds number. However, correlations are unavailable (at least in the open literature) for systems such as the double-wall geometry considered here, where the system is an amalgamation of various cooling features that affect the performance of each other. To create a thermal distribution in the two-dimensional section representative of that which would be expected during operation, a thermal model was developed by Murray et al.  who utilized a combination of empirical correlations and observations informed by computational simulations. The output of this model is the temperature at the external surface of the outer skin and a Nusselt number (and consequently heat transfer coefficient) distribution over the entire internal surface of the representative geometry. The details of the thermal model are given in Secs. 3.1–3.4.
3.1 The External Surface of the Outer Skin.
3.2 The Inner Surface of the Outer Skin.
Local heat transfer coefficients can be most significantly affected by impingement cooling  where the high velocity coolant jet impacts the hot surface resulting in very high stagnation point heat transfer coefficients, with the flow also acting to increase turbulent mixing after impingement. CFD typically struggles to accurately predict impingement jet heat transfer using lower complexity turbulence models , and large eddy simulations, while providing good accuracy, require high computational resources. As a consequence, empirical correlations are typically used in predicting area-averaged or stagnation point Nusselt number based on flow characteristics such as the impingement jet Reynolds number, as well as target plate from impingement hole distance (i.e., H in the given unit cell). In the current model, three correlations were used in predicting impingement zone local Nusselt number. These are as follows:
3.3 The Impingement and Film Holes.
3.4 The Pedestal.
The Nusselt number on all other surfaces was approximated based on conjugate heat transfer CFD simulations that have been performed on double-wall cooling geometries. For these surfaces (the pedestal surfaces and inner skin), high-fidelity CFD simulations were used to predict the surface heat transfer coefficient distributions. The data were then made dimensionless and curves fitted allowing approximate Nusselt number trends to be predicted in geometries not explicitly simulated in CFD. The Nusselt numbers predicted in this way were scaled in differing geometries based on the impingement zone data that were based on empirical correlations. Figure 7 demonstrates one such curve for the distribution in Nusselt number along the length of the pedestal (the side facing the impingement zone). The figure demonstrates the relatively high heat transfer occurring toward the top of the pedestal, a result of the impingement flow striking this section of the pedestal. The trend is very similar to those observed experimentally by Son et al.  for protrusions connected to the surface of the impingement target wall. At very large distances between the impingement zone and the pedestal, this increase in heat transfer was not so prominent; however, at spacings commonly associated with double-wall type cooling geometries, this prediction of Nu distribution is appropriate.
4 Stress Analysis of the Double-Wall Cooling System
In this section, we present the different approaches that are used to analyze the thermomechanical boundary value problem in Sec. 2. We start by analyzing the temperature and the heat transfer coefficient fields as described in Sec. 3 and develop different orders of approximation. A theoretical analysis of the problem is then provided for the first-order approximation. To investigate the influence of the higher order approximations of the temperature and convection coefficient fields, we developed a computational model using the finite element method.
4.1 The Perturbation Analysis of the Thermal Fields.
4.2 The Analysis of the Heat Transfer Coefficient.
The amplitudes of first-order and higher order terms depend on the Reynolds number at the impingement hole Rei and the different dimensions of the unit cell, i.e., Di/t and H/t. Consider the expressions in Eqs. (12)–(15) and the fitting results in Fig. 7, the first-order approximation for the case of impingement hole diameter of Di/t = 0.5, t = 1 mm, and H/t ∈ [3, 10] and different values of Reynolds number are determined and illustrated in Fig. 8. The amplitudes of the higher order approximations are found, using the thermal model in Sec. 3, to be linearly related to the first order by , , and . The resulting shape functions are shown in Fig. 9. The heat transfer coefficient on the inner walls can be characterized by Biot numbers of the first-order approximation and the total field, i.e., and Bik = hkt/K, respectively, where K is the thermal conductivity of the material.
4.3 Theoretical Analysis.
4.4 Numerical Analysis.
The initial-boundary value problem described in Section 2.3 is numerically solved using the FE code abaqus . A coupled thermomechanical analysis is used. The FE analysis requires the solution for an elastic constitutive law and Fourier’s law of heat conduction. Suitable material parameters have been used, i.e., for CMSX-4 nickel-base superalloy, in the analysis that are the thermal conductivity coefficient K = 14 W/m K, the thermal expansion coefficient α = 12 × 10−6 m/m K, Young’s modulus E = 120 GPa, and Poisson ratio ν = 0.3 for the temperature range 25 °C–500 °C [19–21]. The values of convection coefficient are used for a specific range of Reynolds numbers as described in Sec. 3. The geometry of the axisymmetric unit cell shown in Fig. 4 is discretized, and a typical finite element mesh is shown in Fig. 11. The initial dimensions are taken as L = 1 mm, H = 1 mm, t = 1 mm, and Dpd = 1 mm. A parametric study on different geometrical ratios, i.e., L/t, H/t, and Dpd/t, is performed by keeping t constant and changing other dimensions. The four-node reduced integration bilinear axisymmetric elements (CAX4RT) are used in the discretization. The mesh has 2612 elements, which is found to be sufficient to obtain converged solutions. A uniform refined element region is created adjacent to the fillet to accurately obtain the different stresses, see Fig. 11(b). The temperature and heat transfer coefficients are mapped on to the nodes of different surfaces according to Eq. (16). The periodic boundary conditions in Eq. (8) are applied using kinematical coupling. The rotational degree-of-freedom perpendicular to the rz-plane, urz, of the symmetry surface nodes are set to be equal to the master node C as illustrated in Fig. 11(a).
5 Results and Discussion
5.1 The Effect of the Geometrical Features.
Initially, we have studied the relation between the stress and different geometric ratios of the unit cell. In particular, the geometric ratios L/t, H/t, Dpd/t, and R/t are considered. The analysis is limited to the case of first-order approximations of the temperature. (The influence of higher order terms is investigated later.) Before evaluating the computational results in detail, it is instructive to compare the theoretical and numerical models. Hence, the computational model is solved for the case of the Biot number (Rei → ∞) and different values of the geometrical ratio H/t. The Biot number is chosen to be large enough to keep the pedestal and inner skin temperature equal to the coolant temperature. Figure 12 shows the relation between the maximum nominal stress and the ratio H/t. The figure shows that the theoretical and computational model are in excellent agreement. The result indicates that increases with the increase of H/t and approaches its maximum value of 0.5 when H/t → ∞. This ratio determines the constraining of the outer skin as in the periodicity condition in Eq. (30).
Figures 13(a)–13(d) illustrate the stress distributions in the radial, circumferential, and first and third principal directions. The distributions of the radial and circumferential stresses are identical with a larger stress concentration in the case of radial stress at the fillet. The first and third principal stresses are equal to the tensile and compressive radial stresses, respectively. Thus, the results confirm that the stresses are bending stresses, which are generated by the thermal curvature.
The distance between the pedestals, L/t, is now investigated. Figure 14 shows plotted against H/t for Dpd/t = 1, R/t = 0.1, and different values of L/t. The result implies that the maximum nominal stress decreases with the increase of L/t and approaches the theoretical solution in Eq. (36) for L/t ≥ 2. Figures 15(a)–15(d) illustrates the distribution of the nominal stress for different values of L/t, which implies that, in the case of smaller L/t, the nominal stress is influenced by the stress concentration in the fillet. Therefore, the maximum nominal stress becomes dependent on the stress concentration factor of the fillet and the distance L/t. However, the maximum value of the stress in the fillet for the constant values of Dpd/t and R/t used in Fig. 15 appears to be constant for the different values of L/t employed (i.e., ), which can be explained by the fact that the unit cell is subjected to deformation controlled loading conditions.
We now examine the stress concentration factors Kr and Kθ at the fillet using the FE model for different combinations of the fillet radius R/t and pedestal diameter Dpd/t. It should be mentioned that the stress concentration factors are independent of L/t when expressed as a multiple of the maximum nominal stress determined from the theoretical analysis. Figure 16 shows the stress concentration factors as functions of the dimensionless fillet radius R/t and pedestal diameters Dpd/t. The results suggest that the stress concentration factor is larger in the radial direction in comparison with the circumferential direction, and both factors increase with the decrease of R/t and increase of Dpd/t.
5.2 Convection Versus Conduction Heat Transfer.
In order to study the role of convection cooling on the inner walls, several analyses were performed for the first-order approximation of the heat transfer coefficient and different values of Reynolds number Rei = 104, 105, and 106 as in Fig. 8 for Di/t = 0.5. For the given values of Rei, K = 14 W/m K, t = 1 mm, and H/t ∈ [3, 10], the Biot number in the inner walls of the outer skin is determined to be , 90–182, and 514–1030, respectively. The Biot number shows a total change of over the given range of Rei. The temperature at the external surface is taken to be the first-order approximation . Figure 17 shows the relation between and H/t for different values of the Reynolds number. The result implies that the decrease of the Reynolds number (and consequently the Biot number) results in a significant reduction in the nominal stress for the entire range of H/t. At lower values of the Reynolds number, increasing the ratio H/t yields a further reduction in the stresses. The temperature and nominal stress distributions are plotted in Figs. 18 and 19, respectively, for different values of Reynolds number. The temperature distribution suggests that as the Biot number decreases, the temperature is conducted through the pedestal to the inner skin and its gradient across the outer skin decreases significantly. Hence, the nominal stress reduction is mainly attributed to the reduction in the temperature gradient as suggested by Eq. (36). Increasing the ratio H/t reduces the Biot number, which results in more heat conduction through the pedestals and further reduction in stresses. Moreover, the nominal stress distribution in the outer skin is greater than in the inner skin at higher values of the Biot number, i.e., where the larger temperature gradient is taking place. At lower values of the Biot number, the thermal gradient in the inner skin becomes larger in comparison with the outer skin, which leads to a higher stress in the inner skin. The theoretical solution in Eq. (36) yields similar results.
5.3 The Perturbation Analysis.
In this section, we investigate the effect of the higher order approximations of the temperature on the stress field. In particular, as discussed in Sec. 4.1, we examine the second contribution to the stress field in Eq. (17), i.e., , which stems from the higher order terms in Eq. (16).
The temperature field is taken to be the second term of Eqs. (16) and (18), i.e., , and the heat flux to the inner wall is taken to be zero. Several analyses were then performed for different values of the wavelength in the range and H/t ∈ [3–10]. To study the nominal stress, we consider two material points A and B that are at , z = 0.5 H, and z = 0.5(H + t), respectively, as shown in Fig. 11. Figures 20(a) and 20(b) illustrate the relation between and the minimum and maximum values of , and λf/L, respectively, where . Furthermore, the temperature, nominal, and first principal stress distributions are plotted in Figs. 21–23, respectively, for three values of λf/L = 0.1, 1.0, and 10.0. The results imply that the stresses depend strongly on the wavelength. At lower values of λf/L, the nominal stress at point A is compressive (very small), and it is tensile at point B. At intermediate values of λf/L, the nominal stress values at points A and B oscillate. In this range, the temperature oscillates along the external surface of the outer skin, which yields a moderate negative temperature gradient across the thickness, leading to higher stresses at the surface and small stresses far from the surface. At higher values of λf/L, the nominal stress at points A and B decrease to the zero. The stress reduction is due to the homogeneous temperature distribution along the external surface of the outer skin, which yields a very small temperature gradient across the thickness. The maximum absolute values of the nominal stress values at points A and B are and 0.06 at λf/L ≈ 0.45 and 0.1, respectively. It should be mentioned that the maximum stress at point A is due to bending deformation and at point B is due to the surface deformation. The maximum and minimum values of the first principal stress are associated with the maximum and minimum nominal stress, see Figs. 22 and 23. Similar oscillatory behavior is observed in the values of the first principal stress. The maximum value of the first principal stress is at the external surface of the outer skin, i.e., at λf/L ≈ 0.8, r = 0, and z = 0.5(H + t). The minimum value of the first principal stress is in the fillet and at λf/L ≈ 0.15. The film hole distribution associated with λf/L ≈ 10 gives the minimum stress distribution in the unit cell.
5.4 The Heat Transfer Coefficient Analysis.
The higher order approximations of the heat transfer coefficient are now investigated. The total heat transfer coefficient at the inner walls in Eq. (16) and the first-order approximation of the temperature at the external surface of the outer skin are used. Similarly, the model is analyzed for the case of Di/t = 0.5, Rei ∈ [102–104], and H/t ∈ [3–10], see Figs. 8 and 9. Figure 24 shows the relation between the nominal stress and the ratio H/t for different values of the Reynolds number and heat transfer coefficient (the first-order approximation and full field hk). The result shows that reducing the Reynolds number (and consequently, the Biot number) yields a significant reduction in the nominal stress for the entire range of H/t. Furthermore, the differences between the stress values of the full field and the first-order approximation are significantly small, i.e., , which can be explained by the fact that the higher order approximations’ amplitude is smaller than the first-order , i.e., the maximum value of the higher order terms in the inner surface of the outer skin is approximately and that of the first-order term is .
6 Concluding Remarks
In this paper, we have investigated thermal stresses in an infinite flat double-wall cooling system. We have considered an idealized axisymmetric unit cell in which the thermal boundary conditions have been applied in terms of the film temperature on the external surface of the outer skin and the heat transfer coefficients at the inner walls. The mechanical boundary conditions are implemented using periodicity conditions. The thermal fields are obtained using empirical correlations and CFD analysis for a given velocity and temperature of the mainstream, Reynolds number at the impingement hole and coolant temperature. Theoretical solutions are initially developed using first-order approximations of the thermal fields and assuming that the thermal conductivity is limited to the outer skin (i.e., it is valid for larger values of Biot number ). A computational model is then developed to solve Fourier’s law of heat conduction and mechanical equilibrium equations using the commercial finite element code abaqus. This model is then used to study different geometrical features and higher order approximations of the thermal fields. The main findings are summarized as follows:
At high values of the Reynolds number at the impingement hole, i.e., Rei ≥ 105, the thermal curvature due to the thermal gradient in the outer skin causes the stresses in the skins. At lower Reynolds number, heat is conducted through the pedestal to the inner skin, which might result in a larger thermal gradient in the inner skin. Therefore, in these case, the thermal curvature in the inner skin drives the deformation.
As a result of first-order approximations of the film temperature, the maximum principal stress occurs at the fillet between the pedestal and outer skin due to stress concentration at higher values of the Reynolds number. At lower Reynolds number, it might change to the fillet between the pedestal and inner skin. The maximum principal stress occurs at the external surface of the outer skin, i.e., at r = 0 and z = 0.5(H + t), in the case of higher order approximations. Hence, depending on the temperature amplitudes , , and Reynolds number, the maximum principal stress might change its magnitude and location.
Increasing the pedestals height-skin thickness ratio H/t increases the nominal stress, which approaches a constant value, i.e., , as H/t → ∞. The nominal stress decreases with the increase of the distance between pedestals; however, the maximum principal stress remains constant due to the constancy of the thermal curvature. The stress concentration factor at the fillet increases with the increase of the pedestal diameter and decreases with the increase of its radius and skin thickness.
The higher order approximation of the film temperature is characterized by the number and distribution of the film holes and is expressed in terms of the wavelength λf and the phase angle ϕf. Increasing the number of film holes limits stresses to the external surface of the outer skin. Moderate and smaller numbers result in higher and lower stresses, respectively, as the inhomogeneous temperature distribution at the external surface of the outer skin becomes homogeneous.
The higher order approximation of the heat transfer coefficients plays a minor role due to the fact that their amplitudes are less than of the first-order approximation.
These calculations provide a physical insight into geometric factors that determine how stresses develop in double-wall effusion cooled systems during thermal cycling. This understanding can be used to help identify structural configurations, which provide good mechanical performance. This will be addressed in the subsequent work.
The authors are grateful to both Rolls-Royce plc and EPSRC under grant EP/P000878/1 for supporting the research presented in this paper.
heat transfer coefficient (W/(m2 K))
length scale (m)
the outer and inner skin thicknesses (m)
the diameter (m)
Young’s modulus (N/m2)
the height of the pedestals (m)
thermal conductivity (W/(m K))
the spacing between two adjacent pedestals (m)
the moment per unit thickness (N m/m)
the normal force per unit thickness (N/m)
the radius of the fillets between the pedestals and the outer and inner skins (m)
specific heat capacity (J/K)
displacement vector (i = 1, 2, 3)
Nusselt number, Nu = h l/K
Prandtl number, Pr = cpμ/K
Reynolds number, Re = ρv l/μ
the thermal expansion coefficient (K−1)
engineering strain tensor
dynamic viscosity (Pa · s)
Cauchy stress tensor (Pa)