The old idea that an infinite dimensional dynamical system may have its high modes or frequencies... more The old idea that an infinite dimensional dynamical system may have its high modes or frequencies slaved to low modes or frequencies is re-visited in the context of the 3D Navier-Stokes equations. A set of dimensionless frequencies {Ω m (t)} are used which are based on L 2m-norms of the vorticity. To avoid using derivatives a closure is assumed that suggests that theΩ m (m > 1) are slaved toΩ 1 (the global enstrophy) in the formΩ m =Ω 1 F m (Ω 1). This is shaped by the constraint of two Hölder inequalities and a time average from which emerges a form for F m which has been observed in previous numerical Navier-Stokes and MHD simulations. When written as a phase plane in a scaled form, this relation is parametrized by a set of functions 1 λ m (τ) 4, where curves of constant λ m form the boundaries between tongue-shaped regions. In regions where 2.5 λ m 4 and 1 λ m 2 the Navier-Stokes equations are shown to be regular : numerical simulations appear to lie in the latter region. Only in the central region 2 < λ m < 2.5 has no proof of regularity been found.
An investigation of lower bounds on the quantities for for the incompressible three-dimensional (... more An investigation of lower bounds on the quantities for for the incompressible three-dimensional (3D) Euler equations has led us to consider a set of spatially averaged weighted eigenvalues, and , of the strain matrix and the Hessian matrix of the pressure , ...
The 3D incompressible Euler equations with a passive scalar θ are considered in a smooth domain Ω... more The 3D incompressible Euler equations with a passive scalar θ are considered in a smooth domain Ω ⊂ R 3 with no-normal-flow boundary conditions u •n| ∂Ω = 0. It is shown that smooth solutions blow up in a finite time if a null (zero) point develops in the vector B = ∇q × ∇θ, provided B has no null points initially : ω = curl u is the vorticity and q = ω • ∇θ is a potential vorticity. The presence of the passive scalar concentration θ is an essential component of this criterion in detecting the formation of a singularity. The problem is discussed in the light of a kinematic result by Graham and Henyey (2000) on the non-existence of Clebsch potentials in the neighbourhood of null points.
The issue of why computational resolution in Navier–Stokes turbulence is hard to achieve is addre... more The issue of why computational resolution in Navier–Stokes turbulence is hard to achieve is addressed. Under the assumption that the three-dimensional Navier–Stokes equations have a global attractor it is nevertheless shown that solutions can potentially behave differently in two distinct regions of space–time $\mathbb{S}$± where $\mathbb{S}$− is comprised of a union of disjoint space–time ‘anomalies’. If $\mathbb{S}$− is non-empty it is dominated by large values of |∇ω|, which is consistent with the formation of vortex sheets or tightly coiled filaments. The local number of degrees of freedom ± needed to resolve the regions in $\mathbb{S}$± satisfies $\mathcal{N}^{\pm}(\bx,\,t)\lessgtr 3\sqrt{2}\,\mathcal{R}_{u}^{3},$, where u = uL/ν is a Reynolds number dependent on the local velocity field u(x, t).
For the class of cylindrically symmetric velocity fields U(r, z, t) = {u(r, t), v(r, t), zγ (r, t... more For the class of cylindrically symmetric velocity fields U(r, z, t) = {u(r, t), v(r, t), zγ (r, t)}, two infinite energy exact solutions of the three-dimensional incompressible Euler equations are exhibited that blow up at every point in space in finite time. The first solution is embedded within the second as a special case and in both cases v = 0. Both solutions represent three-dimensional vortices which take the form of hollow cylinders for which the vorticity vector is ω = (0, ω θ , 0). An analysis on characteristics shows how more general solutions can be constructed and analysed.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2010
Higher moments of the vorticity field Ω m ( t ) in the form of L 2 m -norms ( ) are used to explo... more Higher moments of the vorticity field Ω m ( t ) in the form of L 2 m -norms ( ) are used to explore the regularity problem for solutions of the three-dimensional incompressible Navier–Stokes equations on the domain . It is found that the set of quantities provide a natural scaling in the problem resulting in a bounded set of time averages 〈 D m 〉 T on a finite interval of time [0, T ]. The behaviour of D m +1 / D m is studied on what are called ‘good’ and ‘bad’ intervals of [0, T ], which are interspersed with junction points (neutral) τ i . For large but finite values of m with large initial data ( Ω m (0)≤ ϖ 0 O ( Gr 4 )), it is found that there is an upper bound which is punctured by infinitesimal gaps or windows in the vertical walls between the good/bad intervals through which solutions may escape. While this result is consistent with that of Leray (Leray 1934 Acta Math. 63 , 193–248 ( doi:10.1007/BF02547354 )) and Scheffer (Scheffer 1976 Pacific J. Math. 66 , 535–552),— this...
One of the outstanding open questions in modern applied mathematics is whether solutions of the i... more One of the outstanding open questions in modern applied mathematics is whether solutions of the incompressible Euler equations develop a singularity in the vorticity field in a finite time. This paper briefly reviews some of the issues concerning this problem, together with some observations that may suggest that it may be more subtle than first thought.
Vorticity dynamics of the three-dimensional incompressible Euler equations are cast into a quater... more Vorticity dynamics of the three-dimensional incompressible Euler equations are cast into a quaternionic representation governed by the Lagrangian evolution of the tetrad consisting of the growth rate and rotation rate of the vorticity. In turn, the Lagrangian evolution of this tetrad is governed by another that depends on the pressure Hessian. Together these form the basis for a direction of vorticity theorem on Lagrangian trajectories. Moreover, in this representation, fluid particles carry ortho-normal frames whose Lagrangian evolution in time are shown to be directly related to the Frenet-Serret equations for a vortex line. The frame dynamics suggest an elegant Lagrangian relation regarding the pressure Hessian tetrad. The equations for ideal MHD are similarly considered.
Applied Analysis of the Navier-Stokes Equations, 1995
Introduction In the previous chapter it was shown how solutions of the Navier-Stokes equations co... more Introduction In the previous chapter it was shown how solutions of the Navier-Stokes equations could be constructed. Uniqueness of those solutions requires more regularity, however, than that which follows directly from their construction via the Galerkin approximations. In this chapter we shall begin to see how much regularity is needed to ensure smoothness of solutions of the Navier-Stokes equations. The minimum requirements can be reached for the 2 d problem, but the problem remains open for the 3 d case. This chapter is devoted to the statement and proof of what will be referred to as the ladder theorem for the Navier-Stokes equations on ω = [0, L ] d with periodic boundary conditions and zero mean, and a discussion of its consequences in both 2 d and 3 d . This will enable us to relate the evolution of a seminorm of solutions of the Navier-Stokes equations, containing a given number of derivatives, to one containing a lower number of derivatives. In sections 6.3 and 6.4, it is shown how the ladder leads to the identification of length scales in the solutions. Subsequently, section 6.5 contains those estimates that can be gleaned via the ladder from the 2 d and 3 d Navier-Stokes equations where no assumptions have been made. Finally, to show how forcing fields can be handled differently from the static spatial forcing f(x) of previous chapters, section 6.6 briefly shows how a ladder may be derived for thermal convection. To derive the ladder theorem, it is necessary to introduce the idea of seminorms which contain derivatives of the velocity field higher than unity.
An upper bound on the mixing efficiency is derived for a passive scalar under the influence of ad... more An upper bound on the mixing efficiency is derived for a passive scalar under the influence of advection and diffusion with a body source. For a given stirring velocity field, the mixing efficiency is measured in terms of an equivalent diffusivity, which is the molecular diffusivity that would be required to achieve the same level of fluctuations in the scalar concentration in the absence of stirring, for the same source distribution. The bound on the equivalent diffusivity depends only on the functional "shape" of both the source and the advecting field. Direct numerical simulations performed for a simple advecting flow to test the bounds are reported.
Moments of the vorticity are used to define and estimate a hierarchy of time-averaged inverse len... more Moments of the vorticity are used to define and estimate a hierarchy of time-averaged inverse length scales for weak solutions of the three-dimensional, incompressible Navier-Stokes equations on a periodic box. The estimate for the smallest of these inverse scales coincides with the inverse Kolmogorov length but thereafter the exponents of the Reynolds number rise rapidly for correspondingly higher moments. The implications of these results for the computational resolution of small scale vortical structures are discussed.
We address the problem in Navier-Stokes isotropic turbulence of why the vorticity accumulates on ... more We address the problem in Navier-Stokes isotropic turbulence of why the vorticity accumulates on thin sets such as quasi-one-dimensional tubes and quasi-two-dimensional sheets. Taking our motivation from the work of Ashurst, Kerstein, Kerr and Gibson, who observed that the vorticity vector ω aligns with the intermediate eigenvector of the strain matrix S, we study this problem in the context of both the three-dimensional Euler and Navier-Stokes equations using the variables α = ˆ ξ ·S ˆ ξ and χ = ˆ ξ ×S ˆ ξ where ˆ ξ = ω/ω. This introduces the, which lies between ω and Sω. For the Euler equations dynamic angle φ(x, t) = arctan ( χ α a closed set of differential equations for α and χ is derived in terms of the Hessian matrix of the pressure P = {p,ij}. For the Navier-Stokes equations, the Burgers vortex and shear layer solutions turn out to be the Lagrangian fixed point solutions of the equivalent (α, χ) equations with a corresponding angle φ = 0. Under certain assumptions for more g...
Moments of the vorticity are used to define and estimate a hierarchy of time-averaged inverse len... more Moments of the vorticity are used to define and estimate a hierarchy of time-averaged inverse length scales for weak solutions of the three-dimensional, incompressible Navier-Stokes equations on a periodic box. The estimate for the smallest of these inverse scales coincides with the inverse Kolmogorov length but thereafter the exponents of the Reynolds number rise rapidly for correspondingly higher moments. The implications of these results for the computational resolution of small scale vortical structures are discussed.
Two possible diagnostics of stretching and folding (S&F) in fluid flows are discussed, based on t... more Two possible diagnostics of stretching and folding (S&F) in fluid flows are discussed, based on the dynamics of the gradient of potential vorticity (q = ω • ∇θ) associated with solutions of the three-dimensional Euler and Navier-Stokes equations. The vector B = ∇q × ∇θ satisfies the same type of stretching and folding equation as that for the vorticity field ω in the incompressible Euler equations (Gibbon & Holm, 2010). The quantity θ may be chosen as the potential temperature for the stratified, rotating Euler/Navier-Stokes equations, or it may play the role of a seeded passive scalar for the Euler equations alone. The first discussion of these S&F-flow diagnostics concerns a numerical test for Euler codes and also includes a connection with the two-dimensional surface quasi-geostrophic equations. The second S&F-flow diagnostic concerns the evolution of the Lamb vector D = ω × u, which is the nonlinearity for Euler's equations apart from the pressure. The curl of the Lamb vector (̟ := curl D) turns out to possess similar stretching and folding properties to that of the B-vector.
Wc k,~ve undcr:aken a study ~,f the co~:~plex Lorenz equations.~; =-- crx + try. "y =(r-- z)... more Wc k,~ve undcr:aken a study ~,f the co~:~plex Lorenz equations.~; =-- crx + try. "y =(r-- z)x-ay, =- bz, l(x*y + xy*). where x and y ace co~aplex and z is real. The complex parameters r and a are defined by r = rl + it,,: a = 1- ie and o " and b are real. Behaviour ~markably different from the real Lo~,~nz model occurs. Only the origin is a fixed point except for the ~pecial case e + ~ = 0. We have been able to determine analytically two critical values of rt, namely r ~ and rb. The orion is a stable fixed point for 0 < r ~ < r~, but for r)> r~, a Hopf bifurcation to a limit cycle occurs. We have an exact analytic solution for this limit cycle which is always stable if tr < b + I. If o> b + I then this limit is only stable in the region r~ ¢ < r ~ < rf~. When r ~> tie, a transiEon to a finite amplitude oscillation about the limit cycle occurs, The nature of this bifurcation is studied in detail by using a multiple time scale analysis to derive the S...
The old idea that an infinite dimensional dynamical system may have its high modes or frequencies... more The old idea that an infinite dimensional dynamical system may have its high modes or frequencies slaved to low modes or frequencies is re-visited in the context of the 3D Navier-Stokes equations. A set of dimensionless frequencies {Ω m (t)} are used which are based on L 2m-norms of the vorticity. To avoid using derivatives a closure is assumed that suggests that theΩ m (m > 1) are slaved toΩ 1 (the global enstrophy) in the formΩ m =Ω 1 F m (Ω 1). This is shaped by the constraint of two Hölder inequalities and a time average from which emerges a form for F m which has been observed in previous numerical Navier-Stokes and MHD simulations. When written as a phase plane in a scaled form, this relation is parametrized by a set of functions 1 λ m (τ) 4, where curves of constant λ m form the boundaries between tongue-shaped regions. In regions where 2.5 λ m 4 and 1 λ m 2 the Navier-Stokes equations are shown to be regular : numerical simulations appear to lie in the latter region. Only in the central region 2 < λ m < 2.5 has no proof of regularity been found.
An investigation of lower bounds on the quantities for for the incompressible three-dimensional (... more An investigation of lower bounds on the quantities for for the incompressible three-dimensional (3D) Euler equations has led us to consider a set of spatially averaged weighted eigenvalues, and , of the strain matrix and the Hessian matrix of the pressure , ...
The 3D incompressible Euler equations with a passive scalar θ are considered in a smooth domain Ω... more The 3D incompressible Euler equations with a passive scalar θ are considered in a smooth domain Ω ⊂ R 3 with no-normal-flow boundary conditions u •n| ∂Ω = 0. It is shown that smooth solutions blow up in a finite time if a null (zero) point develops in the vector B = ∇q × ∇θ, provided B has no null points initially : ω = curl u is the vorticity and q = ω • ∇θ is a potential vorticity. The presence of the passive scalar concentration θ is an essential component of this criterion in detecting the formation of a singularity. The problem is discussed in the light of a kinematic result by Graham and Henyey (2000) on the non-existence of Clebsch potentials in the neighbourhood of null points.
The issue of why computational resolution in Navier–Stokes turbulence is hard to achieve is addre... more The issue of why computational resolution in Navier–Stokes turbulence is hard to achieve is addressed. Under the assumption that the three-dimensional Navier–Stokes equations have a global attractor it is nevertheless shown that solutions can potentially behave differently in two distinct regions of space–time $\mathbb{S}$± where $\mathbb{S}$− is comprised of a union of disjoint space–time ‘anomalies’. If $\mathbb{S}$− is non-empty it is dominated by large values of |∇ω|, which is consistent with the formation of vortex sheets or tightly coiled filaments. The local number of degrees of freedom ± needed to resolve the regions in $\mathbb{S}$± satisfies $\mathcal{N}^{\pm}(\bx,\,t)\lessgtr 3\sqrt{2}\,\mathcal{R}_{u}^{3},$, where u = uL/ν is a Reynolds number dependent on the local velocity field u(x, t).
For the class of cylindrically symmetric velocity fields U(r, z, t) = {u(r, t), v(r, t), zγ (r, t... more For the class of cylindrically symmetric velocity fields U(r, z, t) = {u(r, t), v(r, t), zγ (r, t)}, two infinite energy exact solutions of the three-dimensional incompressible Euler equations are exhibited that blow up at every point in space in finite time. The first solution is embedded within the second as a special case and in both cases v = 0. Both solutions represent three-dimensional vortices which take the form of hollow cylinders for which the vorticity vector is ω = (0, ω θ , 0). An analysis on characteristics shows how more general solutions can be constructed and analysed.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2010
Higher moments of the vorticity field Ω m ( t ) in the form of L 2 m -norms ( ) are used to explo... more Higher moments of the vorticity field Ω m ( t ) in the form of L 2 m -norms ( ) are used to explore the regularity problem for solutions of the three-dimensional incompressible Navier–Stokes equations on the domain . It is found that the set of quantities provide a natural scaling in the problem resulting in a bounded set of time averages 〈 D m 〉 T on a finite interval of time [0, T ]. The behaviour of D m +1 / D m is studied on what are called ‘good’ and ‘bad’ intervals of [0, T ], which are interspersed with junction points (neutral) τ i . For large but finite values of m with large initial data ( Ω m (0)≤ ϖ 0 O ( Gr 4 )), it is found that there is an upper bound which is punctured by infinitesimal gaps or windows in the vertical walls between the good/bad intervals through which solutions may escape. While this result is consistent with that of Leray (Leray 1934 Acta Math. 63 , 193–248 ( doi:10.1007/BF02547354 )) and Scheffer (Scheffer 1976 Pacific J. Math. 66 , 535–552),— this...
One of the outstanding open questions in modern applied mathematics is whether solutions of the i... more One of the outstanding open questions in modern applied mathematics is whether solutions of the incompressible Euler equations develop a singularity in the vorticity field in a finite time. This paper briefly reviews some of the issues concerning this problem, together with some observations that may suggest that it may be more subtle than first thought.
Vorticity dynamics of the three-dimensional incompressible Euler equations are cast into a quater... more Vorticity dynamics of the three-dimensional incompressible Euler equations are cast into a quaternionic representation governed by the Lagrangian evolution of the tetrad consisting of the growth rate and rotation rate of the vorticity. In turn, the Lagrangian evolution of this tetrad is governed by another that depends on the pressure Hessian. Together these form the basis for a direction of vorticity theorem on Lagrangian trajectories. Moreover, in this representation, fluid particles carry ortho-normal frames whose Lagrangian evolution in time are shown to be directly related to the Frenet-Serret equations for a vortex line. The frame dynamics suggest an elegant Lagrangian relation regarding the pressure Hessian tetrad. The equations for ideal MHD are similarly considered.
Applied Analysis of the Navier-Stokes Equations, 1995
Introduction In the previous chapter it was shown how solutions of the Navier-Stokes equations co... more Introduction In the previous chapter it was shown how solutions of the Navier-Stokes equations could be constructed. Uniqueness of those solutions requires more regularity, however, than that which follows directly from their construction via the Galerkin approximations. In this chapter we shall begin to see how much regularity is needed to ensure smoothness of solutions of the Navier-Stokes equations. The minimum requirements can be reached for the 2 d problem, but the problem remains open for the 3 d case. This chapter is devoted to the statement and proof of what will be referred to as the ladder theorem for the Navier-Stokes equations on ω = [0, L ] d with periodic boundary conditions and zero mean, and a discussion of its consequences in both 2 d and 3 d . This will enable us to relate the evolution of a seminorm of solutions of the Navier-Stokes equations, containing a given number of derivatives, to one containing a lower number of derivatives. In sections 6.3 and 6.4, it is shown how the ladder leads to the identification of length scales in the solutions. Subsequently, section 6.5 contains those estimates that can be gleaned via the ladder from the 2 d and 3 d Navier-Stokes equations where no assumptions have been made. Finally, to show how forcing fields can be handled differently from the static spatial forcing f(x) of previous chapters, section 6.6 briefly shows how a ladder may be derived for thermal convection. To derive the ladder theorem, it is necessary to introduce the idea of seminorms which contain derivatives of the velocity field higher than unity.
An upper bound on the mixing efficiency is derived for a passive scalar under the influence of ad... more An upper bound on the mixing efficiency is derived for a passive scalar under the influence of advection and diffusion with a body source. For a given stirring velocity field, the mixing efficiency is measured in terms of an equivalent diffusivity, which is the molecular diffusivity that would be required to achieve the same level of fluctuations in the scalar concentration in the absence of stirring, for the same source distribution. The bound on the equivalent diffusivity depends only on the functional "shape" of both the source and the advecting field. Direct numerical simulations performed for a simple advecting flow to test the bounds are reported.
Moments of the vorticity are used to define and estimate a hierarchy of time-averaged inverse len... more Moments of the vorticity are used to define and estimate a hierarchy of time-averaged inverse length scales for weak solutions of the three-dimensional, incompressible Navier-Stokes equations on a periodic box. The estimate for the smallest of these inverse scales coincides with the inverse Kolmogorov length but thereafter the exponents of the Reynolds number rise rapidly for correspondingly higher moments. The implications of these results for the computational resolution of small scale vortical structures are discussed.
We address the problem in Navier-Stokes isotropic turbulence of why the vorticity accumulates on ... more We address the problem in Navier-Stokes isotropic turbulence of why the vorticity accumulates on thin sets such as quasi-one-dimensional tubes and quasi-two-dimensional sheets. Taking our motivation from the work of Ashurst, Kerstein, Kerr and Gibson, who observed that the vorticity vector ω aligns with the intermediate eigenvector of the strain matrix S, we study this problem in the context of both the three-dimensional Euler and Navier-Stokes equations using the variables α = ˆ ξ ·S ˆ ξ and χ = ˆ ξ ×S ˆ ξ where ˆ ξ = ω/ω. This introduces the, which lies between ω and Sω. For the Euler equations dynamic angle φ(x, t) = arctan ( χ α a closed set of differential equations for α and χ is derived in terms of the Hessian matrix of the pressure P = {p,ij}. For the Navier-Stokes equations, the Burgers vortex and shear layer solutions turn out to be the Lagrangian fixed point solutions of the equivalent (α, χ) equations with a corresponding angle φ = 0. Under certain assumptions for more g...
Moments of the vorticity are used to define and estimate a hierarchy of time-averaged inverse len... more Moments of the vorticity are used to define and estimate a hierarchy of time-averaged inverse length scales for weak solutions of the three-dimensional, incompressible Navier-Stokes equations on a periodic box. The estimate for the smallest of these inverse scales coincides with the inverse Kolmogorov length but thereafter the exponents of the Reynolds number rise rapidly for correspondingly higher moments. The implications of these results for the computational resolution of small scale vortical structures are discussed.
Two possible diagnostics of stretching and folding (S&F) in fluid flows are discussed, based on t... more Two possible diagnostics of stretching and folding (S&F) in fluid flows are discussed, based on the dynamics of the gradient of potential vorticity (q = ω • ∇θ) associated with solutions of the three-dimensional Euler and Navier-Stokes equations. The vector B = ∇q × ∇θ satisfies the same type of stretching and folding equation as that for the vorticity field ω in the incompressible Euler equations (Gibbon & Holm, 2010). The quantity θ may be chosen as the potential temperature for the stratified, rotating Euler/Navier-Stokes equations, or it may play the role of a seeded passive scalar for the Euler equations alone. The first discussion of these S&F-flow diagnostics concerns a numerical test for Euler codes and also includes a connection with the two-dimensional surface quasi-geostrophic equations. The second S&F-flow diagnostic concerns the evolution of the Lamb vector D = ω × u, which is the nonlinearity for Euler's equations apart from the pressure. The curl of the Lamb vector (̟ := curl D) turns out to possess similar stretching and folding properties to that of the B-vector.
Wc k,~ve undcr:aken a study ~,f the co~:~plex Lorenz equations.~; =-- crx + try. "y =(r-- z)... more Wc k,~ve undcr:aken a study ~,f the co~:~plex Lorenz equations.~; =-- crx + try. "y =(r-- z)x-ay, =- bz, l(x*y + xy*). where x and y ace co~aplex and z is real. The complex parameters r and a are defined by r = rl + it,,: a = 1- ie and o " and b are real. Behaviour ~markably different from the real Lo~,~nz model occurs. Only the origin is a fixed point except for the ~pecial case e + ~ = 0. We have been able to determine analytically two critical values of rt, namely r ~ and rb. The orion is a stable fixed point for 0 < r ~ < r~, but for r)> r~, a Hopf bifurcation to a limit cycle occurs. We have an exact analytic solution for this limit cycle which is always stable if tr < b + I. If o> b + I then this limit is only stable in the region r~ ¢ < r ~ < rf~. When r ~> tie, a transiEon to a finite amplitude oscillation about the limit cycle occurs, The nature of this bifurcation is studied in detail by using a multiple time scale analysis to derive the S...
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Papers by J D Gibbon