A parabolic recovery procedure suited for shear stress and heat flux recovery on surfaces from linear element data is proposed. The information required consists of the usual unknowns at points, as well as gradients recovered at the points that are one layer away from the wall. The procedure has been in use for some time and has consistently delivered superior results as compared with the usual wall shear stress and heat flux obtained from linear finite element method shape functions

Appanaboyina, Sunil

Cebral, Juan

Lohner, Rainald

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COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERINGCommun. Numer. Meth. Engng 2008; 24:1611–1615Published online 28 September 2007 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/cnm.1054Parabolic recovery of boundary gradientsRainald Löhner∗,†, Juan Cebral and Sunil AppanaboyinaCenter for Computational Fluid Dynamics, M.S. 6A2, Department of Computational and Data Sciences,George Mason University, Fairfax, VA 22030-4444, U.S.A.SUMMARYA parabolic recovery procedure suited for shear stress and heat flux recovery on surfaces from linearelement data is proposed. The information required consists of the usual unknowns at points, as well asgradients recovered at the points that are one layer away from the wall. The procedure has been in usefor some time and has consistently delivered superior results as compared with the usual wall shear stressand heat flux obtained from linear finite element method shape functions. Copyright q 2007 John Wiley& Sons, Ltd.Received 3 April 2007; Revised 20 August 2007; Accepted 20 August 2007KEY WORDS: higher order gradients; stress recovery; CFD1. BOUNDARY GRADIENT RECOVERYPost-processing results of field solvers, such as those used in computational fluid dynamics (CFD)and computational thermodynamics, often lead to the requirement of evaluating derivatives onboundaries. This is particularly the case if the shear stress or the heat flux on the boundary issought. Many CFD and CSD codes utilize linear elements (triangles, tetrahedral). If we consider atypical boundary element as shown in Figure 1, the most obvious method to evaluate the requiredstress or heat flux is to take the derivatives of the shape functions and obtain the (constant) gradientof the velocities or temperature in the element and multiply it by the material parameters (viscosity,conductivity).Denoting the quantity whose gradient is sought by , and the shape functions as N j , we have,i = xi =Nj,i ̂ j (1)∗Correspondence to: Rainald Löhner, Center for Computational Fluid Dynamics, M.S. 6A2, Department of Compu-tational and Data Sciences, George Mason University, Fairfax, VA 22030-4444, U.S.A.†E-mail: rlohner@gmu.eduCopyright q 2007 John Wiley & Sons, Ltd.1612 R. LÖHNER, J. CEBRAL AND S. APPANABOYINAABC10Figure 1. Element with boundary face A–B–C.Parabolic Recovery0 1 2Φg0g1Linear FEMHermite RecoveryFigure 2. 1-D case.If we consider the 1-D case shown in Figure 2, this leads to,x |0= 1−0h(2)where h is the element size. The recovery of gradients at nodes based on element gradients hasbeen treated before [1]. The basic idea is to represent the gradient of  as a linear function itselfand then obtain the nodal values via a weighted residual method. Suppose that we have gradientinformation at the points inside the domain. In this case, a better approximation to the gradientson the boundary may be obtained using a parabolic extrapolation.With the notation of Figure 2, the function  in the first element may be represented as=0+[21−0h−,x |1]x−[1−0h−,x |1]x2h(3)implying,x |0=21−0h−,x |1 (4)and for 2-/3-D:,n|0=21−0h−,n|1 (5)Copyright q 2007 John Wiley & Sons, Ltd. Commun. Numer. Meth. Engng 2008; 24:1611–1615DOI: 10.1002/cnmPARABOLIC RECOVERY OF BOUNDARY GRADIENTS 1613For the general 3-D case, for each of the boundary faces the following steps are required (seeFigure 1):• obtain the face normal n;• obtain the element adjacent to the face;• obtain the point i1 that is not on the face/surface;• obtain the normal distance h of point i1 to the face;• obtain the value of 0 at the location the line i1, n intersects the face;• obtain the normal derivatives from Equation (5).2. EXAMPLESThe proposed shear stress recovery is shown for two examples. The first is a simple channel withdimensions: length L=5.0, height H =1.0, and width W =0.1. The exact solution is given by theparabola: u=[1−(2y/H)2]u0, where y is the vertical coordinate and u0 the maximum velocity.The values were chosen such that the wall shear stress is =4. Unstructured and structured gridsof different size were generated to observe the convergence to the exact value. Figure 3(a) and (b)shows typical surface grids. Figure 3(b) and (c) shows the shear stress values obtained. Note that, asexpected, the linear finite element method (FEM) gradients under-predict the shear. The parabolic 3.5 3.6 3.7 3.8 3.9 4 4.1 4.2 0.02  0.03  0.04  0.05  0.06  0.07  0.08  0.09  0.1ShearMesh SizeLinFEMParaboExact 3.5 3.6 3.7 3.8 3.9 4 4.1 4.2 0.03  0.04  0.05  0.06  0.07  0.08  0.09  0.1  0.11  0.12  0.13ShearMesh SizeSh_min, LinFEMSh_max, LinFEMSh_ave, LinFEMSh_min, ParaboSh_max, ParaboSh_ave, ParaboExact(a) (b)(c) (d)Figure 3. (a), (b) Surface grids of channel (Cartesian, unstructured) and (c), (d) convergenceof shear stress (Cartesian, unstructured).Copyright q 2007 John Wiley & Sons, Ltd. Commun. Numer. Meth. Engng 2008; 24:1611–1615DOI: 10.1002/cnm1614 R. LÖHNER, J. CEBRAL AND S. APPANABOYINAFigure 4. (a), (b) Aneurysm: surface mesh, surface pressure; (c), (d) aneurysm: shear stress (linear FEM,parabolic recovery); and (e), (f) aneurysm: shear stress (linear FEM, parabolic recovery, RANS mesh).recovery is able to obtain the exact result for the structured grids. For the unstructured grids, somepoints exhibit shear values that are larger or smaller than the exact solution. This is because thegradient recovery at the first point off the wall is performed via linear elements, leading to inexactgradients. Still, the overall accuracy of the parabolic recovery is noticeably higher than that of thelinear FEM.The second example shows the shear stresses obtained from an aneurysm simulation. The mesh,whose surface is displayed in Figure 4(a), had approximately 1.14Mels. The surface pressures aredisplayed in Figure 4(b). The shear stresses obtained using the linear FEM and parabolic recoveryare shown (with the same scale) in Figure 4(c) and (d). Note the marked increase in shear stressrecorded by the parabolic recovery in the neck region.In order to assess grid convergence, a grid was generated that had the same isotropic element sizedistribution as the previous one, but with an additional 10 layers of highly stretched elements inthe wall region to resolve possible boundary layers. The results obtained on this mesh (again, withthe same scale) are displayed in Figure 4(e) and (f). Note that in this case the linear and parabolicstresses are similar, indicating convergence. The run was repeated once more on a globally refinedmesh (with boundary layer gridding) of approximately 10Mels. No discernable differences withthe results shown in Figure 4(e) and (f) were observed, again indicating convergence. Overall,Copyright q 2007 John Wiley & Sons, Ltd. Commun. Numer. Meth. Engng 2008; 24:1611–1615DOI: 10.1002/cnmPARABOLIC RECOVERY OF BOUNDARY GRADIENTS 1615it is remarkable how much better (in absolute terms) the shear stresses obtained from the parabolicrecovery on the isotropic mesh are in comparison with the linear FEM stresses.3. CONCLUSIONSA parabolic recovery procedure suited for shear stress and heat flux recovery on surfaces fromlinear element data has been presented. The information required consists of the usual unknownsat points, as well as gradients recovered at the points that are one layer away from the wall.The procedure has been in use for some time and has consistently delivered superior results ascompared with the usual shear stress obtained from linear FEM shape functions. It may also beused as an error indicator to assess if grid resolution in wall regions is adequate.REFERENCE1. Löhner R. Applied CFD Techniques. Wiley: New York, 2001.Copyright q 2007 John Wiley & Sons, Ltd. Commun. Numer. Meth. Engng 2008; 24:1611–1615DOI: 10.1002/cnm

Parabolic recovery of boundary gradients

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Parabolic recovery of boundary gradients

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