When trying to understand the role and influence of turbulence in our day to day lives, it is useful to imagine what life would be like without it. Turbulent flows are exceptionally good at mixing mass, heat and momentum. We could take numerous industrial or household examples, like how long it would take to mix a cup of coffee without turbulence, however some of the most striking examples relate to how the world around us would be completely different in the absence of turbulent mixing. I cannot remember the source, but I have heard that Kolmogorov used to begin his lectures on turbulence by taking the Volga river in Russia and poseing the question of what the surface velocity of the river would be if there were no turbulence. Taking this as inspiration, let us consider the mean surface velocity of the Yarra river in Melbourne, Australia, if there were no turbulence.
The Yarra river is approximately 240 km long, from a source 790 m above sea level. The mean component of gravitational acceleration acting in the direction of the water flow can be estimated as 0.0033, with a river depth of approximately 4 m as the river passes through Melbourne’s central business district.
The flow in the river can be modelled via the Navier-Stokes equations, which is we consider an incompressible fluid such as water and assuming the average flow to be steady and approximately two dimensional, can be expresses as:
\(U\frac{\partial U}{\partial x} + V\frac{\partial U}{\partial y} = f_x -\frac{1}{\rho}\frac{\partial P}{\partial x} + \nu\left(\frac{\partial^2 U}{\partial x^2} + \frac{\partial^2 U}{\partial y^2}\right)-\frac{\partial \lt u’v’ \gt}{\partial y}\)
where U is the mean velocity in the direction of the river.
Schematic of river coordinate system
I won’t go into detail about each term in this equations as the point of this post is not to explain the fundamental physics at play. As luck would have it, we can neglect the majority of these terms is we assume the river behaves as a fully developed open channel flow such that we can neglect any vertical velocity V and any gradients in the direction of the river flow. This allows us to reduce the equation that governs this flow to:
\( \nu \frac{d^2 U}{d y^2} -\frac{d \lt u’v’ \gt}{d y} + f_x = 0\)
or
\(\nu \frac{d^2 U}{d y^2} -\frac{d \lt u’v’ \gt}{d y} + ig = 0\)
where \(\nu\) is the kinematic viscosity of the water, \(\frac{\partial \lt u’v’ \gt}{\partial y}\) represents the effect of turbulence and ig is the effect of gravity on the flow (component of gravitational force in the flow direction \(g=9.8\) m$\latex ^2$/s).
If we ignore the contribution from turbulence then we are left with a simple ordinary differential equation, which we can solve to determine the velocity of the water at a given depth y.
\(\nu \frac{d^2 U}{d y^2} = -ig\)
Integrating twice we arrive at an equation for the velocity:
\(U(y) = -\frac{ig}{2\nu}y^2+c_1y + c_2\)
where \(c_1\) and \(c_2\) are constants of integration. To determine these constant we need to include two boundary conditions for our river velocity. The first of these is that the velocity at the bottom surface of the river is zero, \(U(0) = 0\) (non-slip condition), the second is that the velocity approaches a constant velocity at the free surface, \(\frac{dU}{dy}(H) = 0\) where H is the river depth or height above the river base.
This result in the following equation for the river velocity:
\(U(y) = \frac{ig}{2\nu}\left(2Hy-y^2\right)\)
in terms of the slope i and depth H of the river and the kinematic viscosity of water, which at these condition is approximately \(\nu = 1\times 10^{-6}\) m\(^2\)/s.
The surface velocity, i.e. U(H) is therefore given by:
\(U(H) = \frac{ig}{2\nu}H^2\)
Returning our focus to the Yarra river, and assuming a mean slope of \(i=0.0033\) and depth \(H=4 \)m, the above equation gives a surface velocity \(U(H) \approx 259\) km/s. This corresponds to nearly a million kilometres per hour, which is clearly ridiculous. While on a positive note this would allow for extremely rapid travel downstream, a river travelling at this velocity would rapidly erode any obstacle placed in its way, and you certainly wouldn’t want to dangle you feet in it on a warm day.
Of course, this doesn’t occur, the reason being that the turbulent term that we neglected clearly plays a significant role in mixing the momentum in the river and ensuring that the river never approaches speeds anywhere near this. Thankfully turbulence is a phenomena that existing at the scales that most influence the world around us. Now, if only we could understand it!