Computational Flow Analysis in Aerospace, Energy and Transportation Technologies with the Variational Multiscale Methods

. With the recent advances in the variational multiscale (VMS) methods, computational (cid:29)ow analysis in aerospace, energy and transportation technologies has reached a high level of sophistication. It is bringing solution in challenging problems such as aerodynamics of parachutes, thermo-(cid:29)uid analysis of ground vehicles and tires, and (cid:29)uid(cid:21)structure interaction (FSI) analysis of wind turbines. The computational challenges include complex geometries, moving boundaries and interfaces, FSI, turbulent (cid:29)ows, rotational (cid:29)ows, and large problem sizes. The Residual-Based VMS (RBVMS), Ar-bitrary Lagrangian-Eulerian VMS (ALE-VMS) and Space(cid:21)Time VMS (ST-VMS) methods have been successfully serving as core methods in addressing the computational challenges. The core methods are supplemented with special methods targeting speci(cid:28)c classes of problems, such as the Slip Interface (SI) method, Multi-Domain Method, and the (cid:16)ST-C(cid:17) data compression method. We provide and overview of the core and special methods. We present, as examples of challenging computations performed with these methods, aerodynamic analysis of a ram-air parachute, thermo-(cid:29)uid analysis of a freight truck and its rear set of tires, and aerodynamic and FSI analysis of two back-to-back wind turbines in atmospheric boundary layer (cid:29)ow.


Introduction
With the recent advances in the variational multiscale (VMS) methods, computational ow analysis in aerospace, energy and transportation technologies has reached a high level of sophistication.
It is bringing solution in many classes of challenging problems. Examples are spacecraft parachute analysis for the landing-stage parachutes [1], cover-separation parachutes [2] and the drogue parachutes [3], spacecraft aerodynamics [2], ram-air parachutes Here, we assume that the tire temperature is given. This assumption is justied because the tire temperature depends on the driving history, which represents a much longer time scale compared to the time scale of the surrounding air.
To make the point, the truck body is 12 m long, and at a driving speed of 80 km/h, a uid particle takes only 0.54 s to travel the full length of the truck. With that, we can decouple the problem into thermo-uid analysis and tire thermostructure analysis. Here we focus only on the thermo-uid part.
In our thermo-uid analysis, the road-surface temperature is higher than the free-stream temperature, and the tire-surface temperature is even higher. The analysis includes the heat from the engine and exhaust system. This is done with a reasonably realistic representation of the rate by which that heat transfer takes place and the surface geometry of the engine and exhaust system over which the heat transfer takes place.
The analysis also includes the heave motion of the truck body, prescribed as a periodic motion with a given semi-amplitude and frequency.

3) Aerodynamic and FSI analysis of two back-to-back HAWTs in turbulent ABL ow
This computation is from [41].
In the momentum equation, ρ, u and f are the density, velocity and body force. The stress tensor σ σ σ(u, p) = −pI + 2µε ε ε(u), where p is the pressure, I is the identity tensor, µ = ρν is the viscosity, ν is the kinematic viscosity, and the strain rate ε ε ε(u) = ∇ ∇ ∇u + (∇ ∇ ∇u) T /2. In the energy equation, C p , θ and κ are the constant-pressure specic heat, temperature and thermal conductivity. In the expression for the body force, β θ , θ ref and a GRAV are the thermal-expansion coefcient, reference temperature and gravitational acceleration. In this mathematical model, ρ and C p are assumed to be constants.
The essential and natural boundary conditions associated with Eq. (1) are represented as h are complementary subsets of the boundary Γ t , n is the unit outward normal vector, and g and h are given functions. The essential and natural boundary conditions associated with Eq.

Structural mechanics
In this article we will not provide any of our formulations requiring uid and structure denitions simultaneously; we will instead give reference to earlier journal articles where the formulations were presented. Therefore, for notation simplicity, we will reuse many of the symbols used in the uid mechanics equations to represent their counterparts in the structural mechanics equations. To begin with, Ω t ⊂ R n sd and Γ t will represent the structure domain and its boundary. The structural mechanics equations are then written, on Ω t and ∀t ∈ (0, T ), as where y and σ σ σ are the displacement and Cauchy stress tensor. The essential and natural boundary conditions for Eq. (7) are represented as y = g on (Γ t ) g and n · σ σ σ = h on (Γ t ) h . The Cauchy stress tensor can be obtained from where F and J are the deformation gradient tensor and its determinant, and S is the second PiolaKirchho stress tensor. It is obtained from the strain-energy density function ϕ as follows: where E is the GreenLagrange strain tensor: and C is the CauchyGreen deformation tensor:  In the ow analyses presented here, we use the YZβ shock-capturing in the thermo-uid analysis of a freight truck and its rear set of tires. We use it for the for the thermal-transport equation.
The DC parameter is a slightly modied version of the one given in [44], which was based on the   [160].  Figure 2 shows one of the ribs.
The suspension lines, used by the parachutist to control the parachute, are modeled as cables.
Fabric patches attached to the parachute sides serve as stabilizers.     The structural mechanics formulation based on the membrane and cable models (see [1]) is supplemented with wrinkling and slacking models (see [164]). The material properties are given in Table 1.
In the computation, we specify the pressure dierence between the two sides of the parafoil surfaces and reel the ends of the suspension lines to the center. Figure 6 shows the pressure dier-      shows the three heat loss locations, which play a role in those estimates.
The local domain contains the left half of the rear tires. Figure 18 shows how it was placed.

Computations and results
The number of nodes and elements in the global and local meshes, which were both generated with tetrahedral elements, are given in Table 2.

ALE computation: aerodynamic and FSI analysis of two back-to-back HAWTs in turbulent ABL ow
This computation is from [41].
In this section, the techniques described are

FSI simulation
In this section we present FSI simulations of the same MDM set-up. The wind-turbine geometry, materials, and mesh, which is comprised of 13,273 quadratic NURBS shell elements, are taken from [41]. Figure 33 shows     [28] Takizawa  "This is an Open Access article distributed under the terms of the Creative Commons Attribution License, 117 which permits unrestricted use, distribution, and reproduction in any medium provided the original work is properly cited (CC BY 4.0)."