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Consider a (Newtonian) incompressible viscous fluid in three spatial dimensions, whose velocity field $\mathbb{v}=\mathbb{v}(x,y,z,t)$ moves according to the Navier-Stokes equations

$$\tag{1}\label{e1}\frac{\partial\mathbb{v}}{\partial t}+(\mathbb{v}\cdot\nabla)\mathbb{v}-\nu\Delta\mathbb{v}=-\nabla p\ , \\ \nabla\cdot\mathbb{v}=0$$

where $\nu$ is the kinematic viscosity and $p$ is the pressure (scalar) field acting on the fluid. Assume that the pressure gradient is always zero: $\nabla p=0$ everywhere for all times $t\geq 0$. Suppose the fluid is in free space (i.e. no boundaries) and we have as an initial condition a smooth velocity field $\mathbb{v}(x,y,z,0)=\mathbb{v}_0(x,y,z)\not\equiv 0$ which vanishes outside a bounded region of $\mathbb{R}^3$.

Question: Is it possible for the flow to develop turbulence in this case?

Edit: as discused in the comments to sammy gerbil's answer below (to whom I thank for helping me make my doubt more precise), my expectation in the absence of a pressure gradient and of boundary drag forces (unlike e.g. in Couette flow between a stationary plate and a moving, parallel one) is that the dissipation term $-\nu\Delta\mathbb{v}$ dominates the convection term $(\mathbb{v}\cdot\nabla)\mathbb{v}$ and the fluid flow should behave like a kind of "heat" flow, dissipating along time until the fluid stops moving (perhaps after an infinite amount of time) - in particular, I expect the flow to remain laminar at all times $t>0$ (hence the tone of the question's title). Put differently, the above question reduces to:

Question (rephrased): Does the linear part of the left hand side of \eqref{e1} (which is essentially a heat operator acting on $\mathbb{v}$) dominate under the above hypotheses?

If that is really true, I would like to see a mathematically precise argument for this, based on the Navier-Stokes equations \eqref{e1}.

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UPDATED ANSWER

Sorry, I interpreted your question too narrowly.

Couette flow occurs without a pressure gradient, due to viscous drag from a boundary surface, and is laminar. If the drag force is increased the flow can become turbulent.

If a transient inertial flow begins laminar I think it must remain laminar as it dies out, because the speed of flow will decrease at all points. (I do not think flow can be laminar at Re1, turbulent at Re2 and laminar again at Re3, where Re1 < Re2 < Re3.)

ORIGINAL ANSWER

For a viscous fluid, if there is no pressure gradient then there is no flow.

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  • $\begingroup$ Even if you start with a non-zero velocity field at $t=0$? (by the way, I should have been more precise about initial conditions - I'll fix that in the question) $\endgroup$ – Pedro Lauridsen Ribeiro Jun 21 '16 at 19:33
  • $\begingroup$ @PedroLauridsenRibeiro : Excluding Couette flow leaves the 2nd option in my answer - transient laminar flow. Do you think turbulence is possible? If so please explain why in your question. $\endgroup$ – sammy gerbil Jun 21 '16 at 20:20
  • $\begingroup$ Regarding Couette flow, you have an external drag force at the boundary, so this should appear in the boundary conditions. Suppose now that the fluid is in free space (i.e. no boundaries) and the initial condition v0 vanishes outside a bounded region of R3 (so there are no boundary drag forces as in Couette flow). Is it still possible to have turbulence? My intuition about this situation is that in this case the fluid flow should behave more like a "heat" flow and dissipate, as you suggested in your updated answer, but I'd like to see a more precise argument. $\endgroup$ – Pedro Lauridsen Ribeiro Jun 21 '16 at 20:20
  • $\begingroup$ I'll edit my question further to include my above expectations, as you suggested. $\endgroup$ – Pedro Lauridsen Ribeiro Jun 21 '16 at 20:21
  • $\begingroup$ @PedroLauridsenRibeiro : Kinetic energy is dissipated even in laminar flow. $\endgroup$ – sammy gerbil Jun 21 '16 at 20:23

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