Numerical approximation of the Voigt regularization for incompressible Navier–Stokes and magnetohydrodynamic flows

Kuberry, Paul; Larios, Adam; Rebholz, Leo G.; Wilson, Nicholas E.

doi:10.1016/j.camwa.2012.07.010

Cited by 35 publications

(24 citation statements)

References 20 publications

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“…Indeed, if one were to add the term (w · ∇)w to the right-hand side of equation (1.3a), the systems would be identical in the f = 0 case. In [32], it was found that in computational tests of the 2D case on a coarse mesh, putting a Voigt term only on (the MHD analogue of) equation (1.3b) resulted in a better match of level curves of the current density (the analogue of ∇ × w) in fine-mesh simulations than putting a Voigt regularization on both equation, or neither equation. This is the reason for us only applying Voigt-regularization to equation (1.3a).…”

Section: Introductionmentioning

confidence: 99%

Global well-posedness of the velocity–vorticity-Voigt model of the 3D Navier–Stokes equations

Larios

Pei

Rebholz

2019

Journal of Differential Equations

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The velocity-vorticity formulation of the 3D Navier-Stokes equations was recently found to give excellent numerical results for flows with strong rotation. In this work, we propose a new regularization of the 3D Navier-Stokes equations, which we call the 3D velocity-vorticity-Voigt (VVV) model, with a Voigt regularization term added to momentum equation in velocity-vorticity form, but with no regularizing term in the vorticity equation. We prove global well-posedness and regularity of this model under periodic boundary conditions. We prove convergence of the model's velocity and vorticity to their counterparts in the 3D Navier-Stokes equations as the Voigt modeling parameter tends to zero. We prove that the curl of the model's velocity converges to the model vorticity (which is solved for directly), as the Voigt modeling parameter tends to zero. Finally, we provide a criterion for finite-time blow-up of the 3D Navier-Stokes equations based on this inviscid regularization.∇ · u = 0, u(·, 0) = u 0 and w(·, 0) = w 0 .where u = (u 1 , u 2 , u 3 ) represents an averaged velocity, w = (w 1 , w 2 , w 3 ), which plays the role of vorticity but for which we do not assume w = ∇ × u, and f is an external forcing term. Without loss of generality, we assume for our analysis in later sections that the viscosity ν = 1. Note that in the case where α = 0, the system formally reduces the velocity-vorticty formulation, while for α > 0, if one imposes w = ∇ × u, the system formally reduces to the Navier-Stokes-Voigt equations.The term −α 2 ∆∂ t u in (1.3a) is often referred to as the "Voigt-term", due to an application of modeling Kelvin-Voigt fluids by A.P. Oskolkov [49,50] (see also [28]). In the context of the velocity formulation use of the Voigt term was first proposed as a regularization for either the Navier-Stokes (for ν > 0) or Euler (for ν = 0) equations in [4], for small values of the regularization parameter α. This paper also proved global well-posedness of the Voigt-regularized versions of the 3D Euler and 3D Navier-Stokes equations.These equations have been studied analytically and extended in a wide variety of contexts (see, e.g.], and the references therein). Voigtregularizations of parabolic equations are a special case of pseudoparabolic equations, that is, equations of the form M u t + N u = f , where M and N are (possibly non-linear, or even non-local) operators. For more about pseudoparabolic equations, see, e.g., [15,52,58,59,57,5,55,56,3].

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Section: Introductionmentioning

confidence: 99%

Global well-posedness of the velocity–vorticity-Voigt model of the 3D Navier–Stokes equations

Larios

Pei

Rebholz

2019

Journal of Differential Equations

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“…This is particularly relevant for the isogrid wall cases because it means that the nature of the boundary layer approaching a subsequent step or discrete obstruction element will be deviated (more turbulent) as compared with a flat-plate boundary layer approaching the element, especially with typical isogrid spacing-to-height (s∕h) ratios of 2 to 8 (see Table 3). CFD simulations of flow over discrete ribs were also completed and shown to be in good agreement with experimental data and low-speed analytical solutions [16][17][18][19].…”

Section: A Experimental Data Collection Of Flow Over Flat Plate and mentioning

confidence: 66%

“…Details of the flow entering into a forward-facing step have been investigated in [16]. The details of the flow structure over discrete rib elements are examined in [17,18], and the flow over a salient fence (essentially a very narrow rib) is studied by researchers in [19].…”

Section: Boundary-layer Modeling Over Walls With Obstruction Elementsmentioning

confidence: 99%

Effect of Isogrid-Type Obstructions on Thermal Stratification in Upper-Stage Rocket Propellant Tanks

Faure

Oliveira

Chintalapati

et al. 2014

Journal of Spacecraft and Rockets

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Analytical models for propellant thermal stratification are typically based on smooth wall flow correlations. However, many propellant tank walls have a mass-saving isogrid, which alters the boundary layer. This work investigates the boundary-layer behavior over walls with obstruction elements that are representative of isogrid or internal stiffener rings. The experimental studies reveal that the thickness of the velocity boundary layer over an isogrid wall is more than 200% thicker than a smooth wall at full-scale upper-stage tank Reynolds numbers. For buoyancy-induced free convection flows, the computational-fluid-dynamics models demonstrate that the velocity boundary layer over a wall lined with obstruction elements may be thicker or thinner than the equivalent boundary layer over a smooth wall, whereas the thermal boundary layer is always thicker for the rough wall. A Rayleigh number scaling analysis is presented for a range of fluids, tank and obstruction sizes, heat loads, and acceleration levels. When the results are applied to a full-scale liquid-hydrogen tank with obstruction elements, the stratification layer is 18% thicker, and the stratum fluid is 31% warmer than the corresponding results for the smooth wall tank. The increase is attributable to the augmented heat transfer area and enhanced mixing of fluid due to the obstruction element. Nomenclature c p = specific heat at constant pressure, J∕kg · K Gr = Grashof number (based on temperature difference) Gr = modified Grashof number (based on heat flux) g = local gravitational acceleration, m∕s 2 H = initial liquid fill level, m h = obstruction height, m L = tank height, m l = length between obstruction, m _ m = mass flow rate, kg∕s N = number of obstruction elements Nu = Nusselt number P = pressure, N∕m 2 Pr = Prandtl number _ q = heat flux, W∕m 2 R = tank radius, m Ra = Rayleigh number (based on temperature difference) Ra = modified Rayleigh number (based on heat flux) Re = Reynolds number S = run length along the obstruction elements, m s = obstruction element center-to-center spacing distance, m t = time, s; or isogrid element thickness, m U, u = velocity, m∕s x, y, z = length, m α = thermal diffusivity, m 2 ∕s β = volumetric thermal expansion coefficient, 1∕K Δ = thickness of stratified layer, m δ = 99% of U e boundary-layer thickness, m δ = boundary-layer displacement thickness, m ε = obstruction parameter ζ = nondimensional velocity η = yU e ∕νx 1∕2 , nondimensional boundary-layer variable θ = temperature difference between wall and fluid, K; or boundary-layer momentum thickness, m κ = thermal conductivity, W∕m · K μ = dynamic viscosity, N · s∕m 2 ν = kinematic viscosity, m 2 ∕s ξ = nondimensional temperature ρ = density, kg∕m 3 τ = implicit time step, s χ = boundary-layer energy flow rate, J∕s ω = spacecraft rotational spin rate, deg ∕s Subscripts b = bulk fluid bl = boundary layer e = external to boundary layer s = stratum td = touchdown distance behind backward-facing step trans = laminar to turbulent boundary-layer transition distance u = ul...

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“…The NS-Voigt equations, and their inviscid counterpart, called the Euler-Voigt equations have been studied analytically and extended in a wide variety of contexts (see, e.g., [6, 8-10, 15, 16, 19, 21, 22, 29, 37, 49, 50, 63, 64, 67, 70, 71, 76, 77, 82, 84, 88, 91-93, 95], and the references therein). Computational studies were carried out in [36,68,74,79]. The NS-Voigt equations were first proposed by Oskolkov in [92,93] as a model for Kelvin-Voigt fluids, but were later viewed as a regularization for the NS equations in [19], where also the Euler-Voigt equations were first introduced and studied.…”

Section: Introductionmentioning

confidence: 99%

Approximate continuous data assimilation of the 2D Navier-Stokes equations via the Voigt-regularization with observable data

Larios¹,

Pei²

2020

Evolution Equations &Amp; Control Theory

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We propose a data assimilation algorithm for the 2D Navier-Stokes equations, based on the Azouani, Olson, and Titi (AOT) algorithm, but applied to the 2D Navier-Stokes-Voigt equations. Adapting the AOT algorithm to regularized versions of Navier-Stokes has been done before, but the innovation of this work is to drive the assimilation equation with observational data, rather than data from a regularized system. We first prove that this new system is globally well-posed. Moreover, we prove that for any admissible initial data, the L 2 and H 1 norms of error are bounded by a constant times a power of the Voigtregularization parameter α > 0, plus a term which decays exponentially fast in time. In particular, the large-time error goes to zero algebraically as α goes to zero. Assuming more smoothness on the initial data and forcing, we also prove similar results for the H 2 norm.

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Numerical approximation of the Voigt regularization for incompressible Navier–Stokes and magnetohydrodynamic flows

Cited by 35 publications

References 20 publications

Global well-posedness of the velocity–vorticity-Voigt model of the 3D Navier–Stokes equations

Global well-posedness of the velocity–vorticity-Voigt model of the 3D Navier–Stokes equations

Effect of Isogrid-Type Obstructions on Thermal Stratification in Upper-Stage Rocket Propellant Tanks

Approximate continuous data assimilation of the 2D Navier-Stokes equations via the Voigt-regularization with observable data

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