Tire aerodynamics with actual tire geometry, road contact and tire deformation
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Abstract
Tire aerodynamics with actual tire geometry, road contact and tire deformation pose tough computational challenges. The challenges include (1) the complexity of an actual tire geometry with longitudinal and transverse grooves, (2) the spin of the tire, (3) maintaining accurate representation of the boundary layers near the tire while being able to deal with the flowdomain topology change created by the road contact and tire deformation, and (4) the turbulent nature of the flow. A new space–time (ST) computational method, “STSITCIGA,” is enabling us to address these challenges. The core component of the STSITCIGA is the ST Variational Multiscale (STVMS) method, and the other key components are the ST Slip Interface (STSI) and ST Topology Change (STTC) methods and the ST Isogeometric Analysis (STIGA). The VMS feature of the STVMS addresses the challenge created by the turbulent nature of the flow, the movingmesh feature of the ST framework enables highresolution flow computation near the moving fluid–solid interfaces, and the higherorder accuracy of the ST framework strengthens both features. The STSI enables movingmesh computation with the tire spinning. The mesh covering the tire spins with it, and the SI between the spinning mesh and the rest of the mesh accurately connects the two sides of the solution. The STTC enables movingmesh computation even with the TC created by the contact between the tire and the road. It deals with the contact while maintaining highresolution flow representation near the tire. Integration of the STSI and STTC enables highresolution representation even though parts of the SI are coinciding with the tire and road surfaces. It also enables dealing with the tire–road contact location change and contact sliding. By integrating the STIGA with the STSI and STTC, in addition to having a more accurate representation of the tire geometry and increased accuracy in the flow solution, the element density in the tire grooves and in the narrow spaces near the contact areas is kept at a reasonable level. We present computations with the STSITCIGA and two models of flow around a rotating tire with road contact and prescribed deformation. One is a simple 2D model for verification purposes, and one is a 3D model with an actual tire geometry and a deformation pattern provided by the tire company. The computations show the effectiveness of the STSITCIGA in tire aerodynamics.
Keywords
Tire aerodynamics Actual tire geometry Road contact ST Variational Multiscale method ST Slip Interface method ST Topology Change method ST Isogeometric Analysis1 Introduction
In this article, we address the computational challenges faced in tire aerodynamics with actual tire geometry, road contact and tire deformation. The article is an updated version of a recent book chapter [1]. The challenges include (1) the complexity of an actual tire geometry with longitudinal and transverse grooves, (2) the spin of the tire, (3) maintaining accurate representation of the boundary layers near the tire while being able to deal with the flowdomain topology change created by the road contact and tire deformation, and (4) the turbulent nature of the flow.
A new space–time (ST) computational method, “STSITCIGA” [2], is enabling us to address the computational challenges. The STSITCIGA was introduced [2] in the context of heart valve flow analysis. Its core component is the ST Variational Multiscale (STVMS) method [3, 4, 5], and the other key components are the ST Slip Interface (STSI) [6, 7] and ST Topology Change (STTC) [8, 9] methods and the ST Isogeometric Analysis (STIGA) [3, 10, 11].
1.1 STVMS
The STVMS is the VMS version of the DeformingSpatialDomain/Stabilized ST (DSD/SST) method [12, 13, 14]. The DSD/SST was introduced for computation of flows with moving boundaries and interfaces (MBI), including fluid–structure interactions (FSI). In MBI computations the DSD/SST functions as a movingmesh method. Moving the fluid mechanics mesh to track an interface enables meshresolution control near the interface and, consequently, highresolution boundarylayer representation near fluid–solid interfaces. The stabilization components of the DSD/SST are the StreamlineUpwind/PetrovGalerkin (SUPG) [15] and PressureStabilizing/PetrovGalerkin (PSPG) [12] stabilizations, which are used very widely. Because of the SUPG and PSPG components, the DSD/SST is now also called “STSUPS.” The VMS components of the STVMS are from the residualbased VMS (RBVMS) method [16, 17, 18, 19]. There are two more stabilization terms beyond those in the STSUPS, and these additional terms give the method better turbulence modeling features. The STSUPS and STVMS, because of the higherorder accuracy of the ST framework (see [3, 4]), are desirable also in computations without MBI.
The Arbitrary Lagrangian–Eulerian (ALE) method is an older and more commonly used movingmesh method. The ALEVMS method [20, 21, 22, 23, 24, 25] is the VMS version of the ALE. It was introduced after the STSUPS [12] and ALESUPS [26] and preceded the STVMS. To increase their scope and accuracy, the ALEVMS and RBVMS are often supplemented with special methods, such as those for weaklyenforced noslip boundary conditions [27, 28, 29], “sliding interfaces” [30, 31] and backflow stabilization [32]. They have been applied to many classes of FSI, MBI and fluid mechanics problems. The classes of problems include windturbine aerodynamics and FSI [33, 34, 35, 36, 37, 38, 39, 40], more specifically, verticalaxis wind turbines [41, 42], floating wind turbines [43], wind turbines in atmospheric boundary layers [44], and fatigue damage in windturbine blades [45], patientspecific cardiovascular fluid mechanics and FSI [20, 46, 47, 48, 49, 50, 51], biomedicaldevice FSI [52, 53, 54, 55, 56, 57], ship hydrodynamics with freesurface flow and fluid–object interaction [58, 59], hydrodynamics and FSI of a hydraulic arresting gear [60, 61], hydrodynamics of tidalstream turbines with freesurface flow [62], and bioinspired FSI for marine propulsion [63, 64].
The STSUPS and STVMS have also been applied to many classes of FSI, MBI and fluid mechanics problems. The classes of problems include spacecraft parachute analysis for the landingstage parachutes [23, 65, 66, 67, 68], coverseparation parachutes [69] and the drogue parachutes [70, 71, 72], windturbine aerodynamics for horizontalaxis windturbine rotors [23, 33, 73, 74], full horizontalaxis windturbines [39, 75, 76, 77] and verticalaxis windturbines [6, 78], flappingwing aerodynamics for an actual locust [10, 23, 79, 80], bioinspired MAVs [76, 77, 81, 82] and wingclapping [8, 83], blood flow analysis of cerebral aneurysms [76, 84], stentblocked aneurysms [84, 85, 86], aortas [87, 88, 89, 90] and heart valves [2, 8, 9, 77, 89, 91, 92], spacecraft aerodynamics [69, 93], thermofluid analysis of ground vehicles and their tires [5, 91], thermofluid analysis of disk brakes [7], flowdriven string dynamics in turbomachinery [94, 95], flow analysis of turbocharger turbines [11, 96, 97, 98], flow around tires with road contact and deformation [1, 91, 99], ramair parachutes [100], and compressibleflow spacecraft parachute aerodynamics [101, 102].
In tireaerodynamics computational analysis, the VMS feature of the STVMS addresses the challenge created by the turbulent nature of the flow, the movingmesh feature of the ST framework enables highresolution flow computation near the moving air–tire interface, and the higherorder accuracy of the ST framework strengthens both features. Furthermore, compared to the tireaerodynamics computational analysis reported in [1], here we use newer element length definitions [103] for the stabilization parameters of the STVMS. The newer definitions are more suitable for isogeometric discretization.
1.2 STSI
The STSI was introduced in [6], in the context of incompressibleflow equations, to retain the desirable movingmesh features of the STVMS and STSUPS when we have spinning solid surfaces, such as a turbine rotor. The mesh covering the spinning surface spins with it, retaining the highresolution representation of the boundary layers. The starting point in the development of the STSI was the version of the ALEVMS for computations with sliding interfaces [30, 31]. Interface terms similar to those in the ALEVMS version are added to the STVMS to account for the compatibility conditions for the velocity and stress at the SI. That accurately connects the two sides of the flow field. An STSI version where the SI is between fluid and solid domains with weaklyenforced Dirichlet boundary conditions for the fluid was also presented in [6]. The SI in this case is a “fluid–solid SI” rather than a standard “fluid–fluid SI.” The STSI method introduced in [7] for the coupled incompressibleflow and thermaltransport equations retains the highresolution representation of the thermofluid boundary layers near spinning solid surfaces. These STSI methods have been applied to aerodynamic analysis of verticalaxis wind turbines [6, 78], thermofluid analysis of disk brakes [7], flowdriven string dynamics in turbomachinery [94, 95], flow analysis of turbocharger turbines [11, 96, 97, 98], flow around tires with road contact and deformation [1, 91, 99], aerodynamic analysis of ramair parachutes [100], and flow analysis of heart valves [2, 89, 92].
In another version of the STSI presented in [6], the SI is between a thin porous structure and the fluid on its two sides. This enables dealing with the fabric porosity in a fashion consistent with how the standard fluid–fluid SIs are dealt with and how the Dirichlet conditions are enforced weakly with fluid–solid SIs. Furthermore, this version enables handling thin structures that have Tjunctions. This method has been applied to incompressibleflow aerodynamic analysis of ramair parachutes with fabric porosity [100]. The compressibleflow STSI methods were introduced in [101], including the version where the SI is between a thin porous structure and the fluid on its two sides. Compressibleflow porosity models were also introduced in [101]. These, together with the compressibleflow ST SUPG method [104], extended the ST computational analysis range to compressibleflow aerodynamics of parachutes with fabric and geometric porosities. That enabled ST computational flow analysis of the Orion spacecraft drogue parachute in the compressibleflow regime [101, 102].
In tireaerodynamics computational analysis, the mesh covering the tire spins with it, and the SI between the spinning mesh and the rest of the mesh accurately connects the two sides of the solution. This enables highresolution representation of the boundary layers near the tire. Furthermore, compared to the tireaerodynamics computational analysis reported in [1], here we use newer element length definitions in [98] for the SI terms of the STSI. The newer definitions are more suitable for isogeometric discretization.
1.3 STTC
The STTC [8, 9] was introduced for movingmesh computation of flow problems with TC, such as contact between solid surfaces. Even before the STTC, the STSUPS and STVMS, when used with robust mesh update methods, have proven effective in flow computations where the solid surfaces are in near contact or create other near TC, if the nearness is sufficiently near for the purpose of solving the problem. Many classes of problems can be solved that way with sufficient accuracy. For examples of such computations, see the references mentioned in [8]. The STTC made movingmesh computations possible even when there is an actual contact between solid surfaces or other TC. By collapsing elements as needed, without changing the connectivity of the “parent” mesh, the STTC can handle an actual TC while maintaining highresolution boundary layer representation near solid surfaces. This enabled successful movingmesh computation of heart valve flows [2, 8, 9, 77, 89, 91, 92], wing clapping [83], and flow around a rotating tire with road contact and prescribed deformation [1, 91, 99].
In tireaerodynamics computational analysis, the STTC enables movingmesh computation even with the TC created by the actual contact between the tire and the road. It deals with the contact while maintaining highresolution flow representation near the tire.
1.4 STSITC
The STSITC is the integration of the STSI and STTC. A fluid–fluid SI requires elements on both sides of the SI. When part of an SI needs to coincide with a solid surface, which happens for example when the solid surfaces on two sides of an SI come into contact or when an SI reaches a solid surface, the elements between the coinciding SI part and the solid surface need to collapse with the STTC mechanism. The collapse switches the SI from fluid–fluid SI to fluid–solid SI. With that, an SI can be a mixture of fluid–fluid and fluid–solid SIs. With the STSITC, the elements collapse and are reborn independent of the nodes representing a solid surface. The STSITC enables highresolution flow representation even when parts of the SI are coinciding with a solid surface. It also enables dealing with contact location change and contact sliding. This was applied to heart valve flow analysis [2, 89, 92] and tire aerodynamics with road contact and deformation [1, 99].
In tireaerodynamics computational analysis, the STSITC enables highresolution flow representation even though parts of the SI are coinciding with the tire and road surfaces. It also enables dealing with tire–road contact location change and contact sliding. Furthermore, compared to the tireaerodynamics computational analysis reported in [1], the newer element length definitions [98] used here for the SI terms of the STSI are more robust in the SITC mechanism, even with finite element discretization.
1.5 STIGA
The STIGA was introduced in [3]. It is the integration of the ST framework with isogeometric discretization. First computations with the STVMS and STIGA were reported in [3] in a 2D context, with IGA basis functions in space for flow past an airfoil, and in both space and time for the advection equation. The stability and accuracy analysis given [3] for the advection equation showed that using higherorder basis functions in time would be essential in getting full benefit out of using higherorder basis functions in space.
In the early stages of the STIGA, the emphasis was on IGA basis functions in time. As pointed out in [3, 4] and demonstrated in [10, 79, 81], higherorder NURBS basis functions in time provide a more accurate representation of the motion of the solid surfaces and a mesh motion consistent with that. They also provide more efficiency in temporal representation of the motion and deformation of the volume meshes, and better efficiency in remeshing. That motivated the development of the ST/NURBS Mesh Update Method (STNMUM) [10, 79, 81]. The name “STNMUM” was given in [75]. The STNMUM has a wide scope that includes spinning solid surfaces. With the spinning motion represented by quadratic NURBS basis functions in time, and with sufficient number of temporal patches for a full rotation, the circular paths are represented exactly, and a “secondary mapping” [3, 4, 10, 23] enables also specifying a constant angular velocity for invariant speeds along the paths. The ST framework and NURBS in time also enable, with the “STC” method, extracting a continuous representation from the computed data and, in largescale computations, efficient data compression [5, 7, 91, 94, 95, 105]. The STNMUM and desirable features of the STIGA with IGA basis functions in time have been demonstrated in many 3D computations. The classes of problems solved are flappingwing aerodynamics for an actual locust [10, 23, 79, 80], bioinspired MAVs [76, 77, 81, 82] and wingclapping [8, 83], separation aerodynamics of spacecraft [69], aerodynamics of horizontalaxis [39, 75, 76, 77] and verticalaxis [6, 78] windturbines, thermofluid analysis of ground vehicles and their tires [5, 91], thermofluid analysis of disk brakes [7], flowdriven string dynamics in turbomachinery [94, 95], and flow analysis of turbocharger turbines [11, 96, 97, 98].
The STIGA with IGA basis functions in space have been utilized in ST computational flow analysis of turbocharger turbines [11, 96, 97, 98], flowdriven string dynamics in turbomachinery [95], ramair parachutes [100], spacecraft parachutes [102], aortas [89, 90], heart valves [2, 89, 92], and tires with road contact and deformation [1]. Most of these computations were accomplished with the integration of the STIGA and STSI or STIGA, STSI and STTC.
1.6 STSITCIGA
The turbocharger turbine analysis [11, 96, 97, 98] and flowdriven string dynamics in turbomachinery [95] were based on the integration of the STSI and STIGA. The IGA basis functions were used in the spatial discretization of the fluid mechanics equations and also in the temporal representation of the rotor and spinningmesh motion. That enabled accurate representation of the turbine geometry and rotor motion and increased accuracy in the flow solution. The IGA basis functions were used also in the spatial discretization of the string structural dynamics equations. This enabled increased accuracy in the structural dynamics solution, as well as smoothness in the string shape and fluid dynamics forces computed on the string.
The ramair parachute analysis [100] and spacecraft parachute compressibleflow analysis [102] were based on the integration of the STIGA, the STSI version that weakly enforces the Dirichlet conditions, and the STSI version that accounts for the porosity of a thin structure. The STIGA with IGA basis functions in space enabled, with relatively few number of unknowns, accurate representation of the parafoil and parachute geometries and increased accuracy in the flow solution. The volume mesh needed to be generated both inside and outside the parafoil. Mesh generation inside was challenging near the trailing edge because of the narrowing space. The spacecraft parachute has a very complex geometry, including gores and gaps. Using IGA basis functions addressed those challenges and still kept the element density near the trailing edge of the parafoil and around the spacecraft parachute at a reasonable level.
The heart valve analysis [2, 89, 92] was based on the integration of the STSI, STTC and STIGA. The STSITCIGA, beyond enabling a more accurate representation of the geometry and increased accuracy in the flow solution, kept the element density in the narrow spaces near the contact areas at a reasonable level. When solid surfaces come into contact, the elements between the surface and the SI collapse. Before the elements collapse, the boundaries could be curved and rather complex, and the narrow spaces might have highaspectratio elements. With NURBS elements, it was possible to deal with such adverse conditions rather effectively.
An SI provides mesh generation flexibility in a general context by accurately connecting the two sides of the solution computed over nonmatching meshes. This type of mesh generation flexibility is especially valuable in complexgeometry flow computations with isogeometric discretization, removing the matching requirement between the NURBS patches without loss of accuracy. This feature was used in the flow analysis of heart valves [2, 89, 92], turbocharger turbines [11, 96, 97, 98], and spacecraft parachute compressibleflow analysis [102].
In tireaerodynamics computational analysis, the STSITCIGA enables a more accurate representation of the geometry and motion of the tire surfaces, a mesh motion consistent with that, and increased accuracy in the flow solution. It also keeps the element density in the tire grooves and in the narrow spaces near the contact areas at a reasonable level. In addition, we benefit from the mesh generation flexibility provided by using SIs.
1.7 Tire models
We present computations with the STSITCIGA and two models of flow around a rotating tire with road contact and prescribed deformation. One is a simple 2D model for verification purposes, and one is a 3D model with an actual tire geometry and a deformation pattern provided by the tire company.
1.8 Outline of the remaining sections
In Sect. 2 we describe the STVMS and STSI. The STSITCIGA is described in Sect. 3. The computations with the 2D and 3D models are presented in Sects. 4 and 5, and the concluding remarks are given in Sect. 6.
2 STVMS and STSI
For completeness, we include, mostly from [6, 99], the STVMS and STSI methods.
2.1 STVMS
Remark 1
The STSUPS can be obtained from the STVMS by dropping the eighth and ninth integrations.
2.2 STSI
2.2.1 Twoside formulation (fluid–fluid SI)
2.2.2 Oneside formulation (fluid–solid SI)
3 STSITCIGA
For completeness, we include (1) from [2, 99] the aspects of the STSI [6] and STTC [8] related to their integration as the STSITC [99] and the advantages of the IGA in this context, and (2) from [2] the integration of all three components as the STSITCIGA.
3.1 STSI
The STSI allows mesh slipping also in the oneside formulation, that is, when the SI serves the purpose of weak enforcement of the Dirichlet boundary conditions for the fluid. The boundary terms added to Eq. (1) to connect the two sides in the fluid–fluid SI and to connect the fluid to the solid in the fluid–solid SI were given in Sects. 2.2.1 and 2.2.2. The added terms [see Eqs. (10) and (17)] include derivatives in the direction normal to the SI. Therefore the elements bordering the SI need to have finite thickness in the normal direction. This places a limitation on the meshes that can be used with the STSI; elements bordering the SI cannot have zero thickness in the normal direction when they degenerate.
3.2 STTC
The STTC can deal with TC in ST movingmesh computations. The discretization is unstructured in time, but based on a parent ST mesh that is structured in time, and the parent mesh is extruded from a single spatial mesh. The key technology in the STTC is massive element degeneration by using a special master–slave system. The special system allows changing, in an ST slab, master nodes to slave nodes and slave nodes to master nodes. With that, elements can collapse or be reborn. This way, in an ST slab, we can represent closing and opening motions. Since an ST method naturally allows discretizations that are unstructured in time, no further modification is needed. With the STTC, we have a method that is very flexible, and computationally as effective as a typical movingmesh method. However, the master–slave relationship has to be node to node; a point on a solid surface that is not a node cannot be a master or slave node.
3.3 STIGA
With NURBS meshes, we can represent curved boundaries with less elements compared to finite element meshes. With this desirable feature, a volume can be meshed also with highaspectratio elements. This is particularly helpful when we need to generate meshes in very narrow spaces.
3.4 STSITCIGA
Integration of the STSI, STTC and STIGA brings several good features to ST computations. (1) It enables highresolution boundary layer representation near the solid surfaces in contact even when the surfaces are covered by meshes with SI. (2) It enables dealing with contact location change and contact sliding on the SI. This overcomes the STTC restriction created by the rule that a point on a solid surface that is not a node cannot be a master or slave node. (3) When part of an SI needs to coincide with a solid surface, which happens for example when the solid surfaces on two sides of an SI come into contact or when an SI reaches a solid surface, the elements between the coinciding SI part and the solid surface need to collapse with the STTC mechanism. Before the elements collapse, the boundaries could be curved and complex, and the narrow space might have highaspectratio elements. With NURBS elements, we can deal with such adverse conditions rather effectively.

If \({h}_\mathrm {B} = 0\), we disregard the integration point, regardless of the value of \({h}_\mathrm {A}\).

If \({h}_\mathrm {B} > 0\) and \({h}_\mathrm {A} = 0\), we use the oneside formulation.

In other cases, we use the twoside formulation.
We note that the SI has a high curvature where it meets the planar surface. To improve the geometric match between the two sides of the SI there, we limit the motion of one of the control points. This in turn reduces the motion of the control points nearby. We periodically remove that limitation for a very short duration, resulting in a sudden jump in the positions of those control points, as can bee seen in the 4th and 5th frames of Figs. 3 and 4.
Remark 2
A node on an SI coinciding with a solid surface must be a slave of the corresponding node on that solid surface.
Remark 3
When for all integration points of an element surface (element edge in the context of the 2D examples) \({h}_\mathrm {B}\) = \({h}_\mathrm {A}\) = 0, that surface is a contact surface. Pressure is not treated as an unknown at a solidsurface master node whose all slave SI nodes live only on contact surfaces. That node has no role in the equation system beyond representing the geometry. Consequently, mesh resolution plays no role in regions made of only contact surfaces. For that reason, in Fig. 4, the stationary mesh in the contact area has very few elements.
4 Verification with a simple 2D model
In this problem a nonmoving mesh can be used to obtain the solution. That will be the reference solution we will compare the STSITCIGA solution to for verification purposes. We will also conduct a verification study by comparing the solutions from two meshes with different refinements.
4.1 Problem setup
4.2 Computational domain, boundary conditions and meshes
4.3 Computational conditions
4.4 Results
Figures 11 and 12 show the velocity magnitude from the preliminarymesh computations with the nonmovingmesh (STSIIGA) and STSITCIGA methods. Overall, the results from the two computations are very comparable.
Figure 13 shows the horizontal component of the flow velocity computed with the nonmovingmesh (STSIIGA) method, using the preliminary and refined meshes. The spinning surface generates a flow relative to the planar surface, creating boundary layers near the spinning and planar surfaces. The preliminarymesh solution has just slightly more fluctuations than the refinedmesh solution, and we can see convergence.
To compare the solutions obtained with the STSITCIGA and nonmovingmesh (STSIIGA) methods, Figs. 14 and 15 show the horizontal component of the flow velocity computed with these two methods, using the preliminary and refined meshes. The solutions obtained with the two methods are in close agreement, indicating that the STSITCIGA method can accurately represent the boundary layers in this class of flow problems, including the boundary layers in regions near the contact.
5 Tire aerodynamics with an actual tire geometry
5.1 Problem setup
5.2 Computational domain, boundary conditions and meshes
The computational domain is shown in Fig. 19. The domain size is \(4.000~\text {m}\) and \(5.489~\text {m}\) in width and height, and \(8.000~\text {m}\) in the flow direction. The tire is placed at \(2.000~\text {m}\) from the inflow boundary. The boundary conditions are 3D extensions of the conditions in the simple 2D model, with slip conditions on the boundary planes perpendicular to the tire axis. We use two different quadratic NURBS meshes: a preliminary mesh and a refined mesh. The number of control points and elements for the two meshes are given in Table 1.
Figure 20 shows, for the two meshes, the refinement level near the tire surface. As can be discerned from the figure, the refined mesh has twice the resolution in the normal direction, and four times the resolution in the longitudinal direction. In the transverse direction, it has four times the resolution across the treads, and twice the resolution across the grooves. We note that most of the mesh generation complexity is near the tire surfaces, which we wanted to do manually. The rest of the mesh could have been generated automatically (see [96]), but that was also generated manually, because it was not that difficult.
5.3 Computational conditions
Tire aerodynamics with an actual tire geometry. Number of control points (nc) and elements (ne) for the two quadratic NURBS meshes used in the computations
Preliminary  Refined  

nc  690,144  4,149,720 
ne  376,560  2,921,552 
5.4 Results
Figures 21 and 22 show, for the two meshes, the velocity magnitude near the contact area. In the solution obtained with the preliminary mesh, the flow patterns are closer to the tire surface. Figures 23 and 24 show, for the two meshes, the isosurfaces corresponding to a positive value of the second invariant of the velocity gradient tensor, colored by the velocity magnitude. The solution obtained with the refined mesh has a better resolution of the vortex structure. This confirms the importance of having a good method and high resolution near the tire–road contact areas.
6 Concluding remarks
We have successfully addressed the computational challenges faced in tire aerodynamics with actual geometry, road contact and tire deformation. The challenges include (1) the complexity of an actual tire geometry with longitudinal and transverse grooves, (2) the spin of the tire, (3) maintaining accurate representation of the boundary layers near the tire while being able to deal with the flowdomain topology change created by the road contact and tire deformation, and (4) the turbulent nature of the flow. The STSITCIGA, a new ST computational method, has enabled us to overcome these challenges. The core component of the STSITCIGA is the STVMS, and the other key components are the STSI, STTC and STIGA. The challenge created by the turbulent nature of the flow is addressed with the VMS feature of the STVMS. The movingmesh feature of the ST framework enables highresolution flow computation near the air–tire interfaces as the tire rotates. These two features are enhanced with the higherorder accuracy of the ST framework. With the STSI, we are able to do movingmesh computations with the tire spinning. The mesh covering the tire spins with it, and the SI between the spinning mesh and the rest of the mesh accurately connects the two sides of the solution. With the STTC, we are able to do movingmesh computations even with the TC created by the contact between the tire and the road. This enables dealing with the contact while maintaining highresolution flow representation near the tire. Integration of the STSI and STTC enables highresolution flow representation even though parts of the SI are coinciding with the tire and road surfaces. It also enables dealing with the tire–road contact location change and contact sliding. Integration of the STIGA with the STSI and STTC not only enables a more accurate representation of the tire geometry and increased accuracy in the flow solution, but also keeps the element density in the tire grooves and in the narrow spaces near the contact areas at a reasonable level. We presented computations with two models of flow around a rotating tire with road contact and prescribed deformation. One is a simple 2D model for verification purposes, and one is a 3D model with an actual tire geometry and a deformation pattern provided by the tire company. The 2D computations confirm the reliability of the movingmesh and TC features of the STSITCIGA. The 3D computations confirm the importance of having a good method and high resolution near the tire–road contact areas. Overall, the computations show the effectiveness of the STSITCIGA in tire aerodynamics.
Notes
Acknowledgements
This work was supported (first and second authors) in part by GrantinAid for Challenging Exploratory Research 16K13779 from JSPS; GrantinAid for Scientific Research (S) 26220002 from the Ministry of Education, Culture, Sports, Science and Technology of Japan (MEXT); and Rice–Waseda research agreement. This work was also supported (first author) in part by GrantinAid for JSPS Research Fellow 17J10893. The computational method parts of the work were also supported (third author) in part by ARO Grant W911NF1710046 and Top Global University Project of Waseda University. The tire deformation used in Sect. 5 was provided by Bridgestone.
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