Analysis of heat transfer on turbulence-generating ribs using dynamic mode decomposition

Highlights

• Introduces general method for finding a DMD mode which affects heat transfer.

• LES used to extract DMD modes on a gas turbine cooling channel geometry.

• CFD volume mesh morphed to excite or preturb these modes.

• RANS CFD simulation not able to capture any engineering benefit from the morphed meshes.

Abstract

Ducts with turbulence-promoting ribs are common in heat transfer applications and can often found in the form of serpentine channels in gas turbine blades. This study uses a recent modal extraction technique called Dynamic Mode Decomposition (DMD) to determine mode shapes of the spatially and temporally complex flowfield inside a square ribbed (45 degree) duct at Reynolds number 32,400. One subject missing from current literature is a method of directly linking a mode to a certain engineering quantity of interest. Presented is a generalized methodology for producing such a link utilizing the data from the DMD analysis. Exciting the modes which are identified may cause the flow to change in such a way to promote the quantity of interest, in this case, heat transfer. This theory is tested by contouring the walls of the duct using the extracted mode shapes.

An initial, unmodified geometry provides a baseline for comparison to later contoured models. The initial case is run as a steady-state RANS model. LES generates the necessary data for the DMD analysis. Several mode shapes extracted from the flow are applied to the duct walls and run again in the RANS model, then compared to the baseline, and their relative performance examined.

Introduction

In an ever-present quest for higher efficiency, modern gas turbines face increasing firing temperatures, which are imposed on the machine’s turbine. Turbines survive this heat through numerous techniques, one being internal duct cooing (IDC). The internal duct can come in many shapes and sizes, and often includes turbulence-generating ribs (turbulators) to enhance gas mixing near the boundary layer, increasing heat transfer performance at the expensive of compressor bleed pressure. Through most of current gas turbine research, this improvement is found via parametric studies, i.e. running a sweep of a few independent dimensions on a geometry and determining what works well. Han et al. [5] performed many studies on a geometry similar to the one investigated here. This involved simple transverse ribs at varying angles, finding 30–45 degrees generally match or outperform simple 90 degree ribs in heat transfer and pressure loss. Rib geometries can become more complex in V, W, and M arrangements as shown in Singh et al. [16], Kunstmann et al. [7], Write et al. [19], and Dritselis [3]. According to Singh et al. [16], ribs with a V shape perform well across many Reynolds numbers, giving Nusselt ratio $\mathit{Nu}/{\mathit{Nu}}_0$ enhancements over a smooth duct between 2.5 and 3. Another example of an extensive parametric study is by Park et al. [9], whom studied modification of channels with varying aspect ratios, rib heights, and patterns, in some manner ‘assembling’ features of above studies into one.

A different method to improving heat transfer might take a more localized, or high-dimensional parameter, approach where the surface of a duct and rib are modified locally in thousands or millions of locations. With advancements in additive manufacturing such geometries are possible to build. There must then be some way of generating these non-trivial geometries. One way may be to identify fluid mode shapes which are sensitive to heat transfer and modify a duct shape to enhance these modes. This idea was explored using Proper Orthogonal Decomposition by Schwanen and Duggleby [15]. That study found and extracted a POD mode which was most relevant to heat transfer and contoured the walls of a pin-fin array to increase heat transfer. The method introduced is fairly customized to functioning only for heat transfer on appropriate geometries; a more general method is one goal of the present work. While there are many works present in literature involving POD, Ref. [15] is unique in its use of surface deformation to achieve an engineering goal. Another important aspect of mode decomposition methods (though not as engineering-oriented) is the identification of influential flow structures. A study by Park et al. [10] identified important energy-containing structures in an oscillating jet.

The recent introduction of Dynamic Mode Decomposition (DMD) can yield structures (modes) that better describe the transient nature of the turbulent duct flow. DMD was introduced by Schmid [14] and seeks an approximation of the eigenvalues and eigenvectors of transient system’s linear regression. It has been utilized for several studies related to heat transfer; Kalghatgi and Acharya [6] showed a DMD analysis on an inclined cooling jet, Wen et al. [17] decomposed a film cooling jet, and Rowley et al. [12] used DMD on a jet in a crossflow. Another interesting work is Wenwu et al. [18], who used DMD on a film effectiveness field directly, as opposed to utilizing it on a primitive (velocity, temperature, etc) as many studies do. Li et al. [8] performed DMD on various oscillating heated fins and were able to show that the first mode was heavily correlated with heat transfer performance. These studies show various applications of DMD, but do not have a concrete way to describe and quantify which modes are most important to some quantity of interest (in this case, heat transfer).

The current study introduces a possible (and simple) method to rank modes in their contribution to heat transfer and modifies the duct geometry based on those modes. The method is general enough to apply to other engineering quantities such as drag, provided the necessary quantities are used for the DMD. It is applied to a square duct with 45 degree ribs at a Reynolds number of 32400. To the author’s knowledge a DMD analysis on such a geometry represents new ground in the research community. To verify the functionality of the above method, four steps are required; (1) an initial LES simulation to generate needed transient data, (2) a DMD analysis of the LES data, determining modes important to heat transfer, (3) deformation of the mesh to enhance the important modes, and (4) re-execution of LES on the deformed grid, showing amplification of the desired modes and of heat transfer. Due to computational power constraints, step 4 is replaced with several RANS runs. One run is the unmodified geometry, identical to that of the LES case, intended as a baseline for comparison. The remaining runs carry the deformed grid and attempt to show the improvement from the mode decomposition. A flowchart of the proposed process here is shown in Fig. 1. Performing two LES runs would not serve industry well, were LES sees little adoption due to its high expense - therefore this study would also serve to see what can be gained by replacing the second LES run with RANS. RANS is far more attractive for general engineering usage. Large differences in physics between LES and RANS require that caution be used in interpreting meaning behind any changes, though engineering quantities show similarity. At this time verification of the mode ranking methodology is therefore limited.