Is turbulence in air/fluid a random or a deterministic process, and why is it so hard to solve? For sure if it is random it can't be solved...

Or maybe it just appears to be random because of our lack of knowledge to "describe" all initial conditions?

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    $\begingroup$ This question is more suited for Physics.SE. $\endgroup$
    – Bianfable
    Oct 4 at 7:20
  • $\begingroup$ @Bianfable They don't have CFD tag $\endgroup$
    – user71157
    Oct 4 at 7:44
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    $\begingroup$ is it a philosophical question? Practically it doesn't matter: we can simulate it, and we use such simulations to check stall speeds, etc. (in such case we care about "turbulence" not exact if the direction of force at every instant. Physically: it is still classical physics, so just "solve" the Navier-Stokes equation (impossible task but for few cases). Still we use "chaos theory" (or perturbation theories): practically it seems a random process, so we work as it is random. Note: on laminar fluids we still use randomness: pressure and ytemperature are statistical phenomenae. $\endgroup$ Oct 4 at 7:56
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    $\begingroup$ @mins Chaos and randomenss are not same thing. Turbulence is chaos, small change in initial condition completely change output , so called butterfly effect. Chaos is deterministic. $\endgroup$
    – user628075
    Oct 4 at 12:23
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    $\begingroup$ You can still ask questions about CFD on the Physics site. If you can't create a cfd tag someone will do it for you. $\endgroup$ Oct 4 at 22:41

3 Answers 3


The equations that describe fluid flow, the Navier Stokes equations, are generally considered chaotic. In particular turbulence is defined as " fluid motion characterized by chaotic changes in pressure and flow velocity". This means that even when the behaviour and characteristics of the phenomenon (the fluid) are entirely deterministic, the equations still give rise to chaotic behaviour.

What this means is that even when the behaviour of a system is completely deterministic, very small uncertainties in the initial conditions of a fluid can give rise to very large differences in its behaviour. With simple solid mechanics, such as the trajectory of a ball under gravity, a small uncertainty in the initial velocity of a ball gives rise to only a small uncertainty in the final trajectory of the ball. With fluids a small uncertainty in the initial velocity of a fluid, and even very small uncertainties in a calculation, can give rise to a very large difference in the answer. This is what makes turbulent flow problems hard to solve. You need extreme accuracy, extremely low errors and thus extreme computing power in order to get even reasonable accurate solutions to turbulent flow problems.

This behaviour is the case even when the behaviour of the fluid is completely deterministic. There is no need to invoke quantum or other random processes in order to see the unpredictable "chaotic" behaviour of fluids. While quantum effects do technically act on fluids, like on all material things, their effects are far too small to have a noticeable effect on fluid flow predictions, just like they are too small to affect solid mechanics. They are not considered in regular calculations.

TLDR: Fluid flow equations are deterministic (i.e. not random) but are chaotic. This is what makes them hard to solve. Quantum or other random processes have no significant effect.

  • $\begingroup$ Aren't chaotic systems simulated not by increasing accuracy, but rather by running the simulation many times with random initial values and taking the average (simplified) of the results? $\endgroup$
    – Bergi
    Oct 4 at 17:35
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    $\begingroup$ @Bergi No. Stochastic simulation is used for processes that have a random element. $\endgroup$ Oct 4 at 17:41
  • $\begingroup$ Is stochastic simulation a single run process, and if so, does the method itself consist of multithread calculations that sum up to the result shown as a single run simulation? Sorry for possibly inproper terminology :) $\endgroup$
    – Jpe61
    Oct 5 at 9:59
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    $\begingroup$ @DJClayworth "With solid mechanics, such as the trajectory of a ball under gravity, a small uncertainty in the initial velocity of a ball gives rise to only a small uncertainty in the final trajectory of the ball", this is not necessarily true. Some solid mechanics problems like the Three body problem are also chaotic $\endgroup$
    – ROIMaison
    Oct 5 at 10:30
  • $\begingroup$ @ROIMaison That's true, but I'm trying not to complicate the explanation too much. $\endgroup$ Oct 5 at 12:42

We can simulate flow and turbulence in a deterministic way, but only to a certain extent / resolution. For most intents and purposes the accuracy of CFD is quite sufficient, but we will never have enough computing power to calculate flows in accuracy that would fully replicate the actual flow as it happens in nature.

The problem is that flow is atomic level interaction, and to fully simulate or replicate a flow event in a deterministic way, every single variable must be exactly the same for the end result to be exactly the same. This is basically ruined by the underlying random nature of our universe (google virtual particles).

As an aside, I would assert that any process, not just fluid dynamics, we describe as deterministic is that only at "crude enough" resolution.

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    $\begingroup$ Single events are totally random. However, many processes appear deterministic at our level of comprehension. $\endgroup$
    – Jpe61
    Oct 4 at 10:56
  • $\begingroup$ No, I meant single particle level events are random. They all add up to a mostly predictable universe. $\endgroup$
    – Jpe61
    Oct 4 at 11:41
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    $\begingroup$ I'd add Heisenberg's uncertainty principle to the big (or atomically small) picture $\endgroup$
    – sophit
    Oct 4 at 12:05
  • $\begingroup$ It's not quantum effects that make us unable to completely predict fluid flow, it's the inherent chaotic properties of the Navier Stokes equations. $\endgroup$ Oct 4 at 15:44
  • $\begingroup$ Equations do not define anything. They describe. $\endgroup$
    – Jpe61
    Oct 4 at 16:45

As DJClayworth already stated, chaotic systems are deterministic. The outcomes are very sensitive to the initial conditions. Turbulence is a deterministic process. However, the process has partially random outcomes. It would not be correct to say turbulence is free of randomness.

Or maybe just appear to be random because our lack of knowledge to "describe" all initial conditions?

We lack the ability to measure initial conditions exactly.

If we described many of the possible initial conditions and then calculate the outcomes of a deterministic chaotic process, then we will find that the outcomes are partially random.

The randomness is real.

For sure if it is random can't be solved...

"If it is random" does not tell you if a useful physical model exists or not. There are plenty of things that have no known randomness but cause problems for math.

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    $\begingroup$ Also things that have lots of randomness, like motion of gas particles, but very well understood behavior despite that. $\endgroup$ Oct 5 at 20:20

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