12  Bernoulli’s principle and vorticity

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Solving Euler’s equations is hard. It is therefore worthwhile to first try to derive consequences of the equations without actually solving them. So in this lecture we will use our vector calculus skills to manipulate Euler’s equations a bit until they reveal to us Bernoulli’s principle and the vorticity equation.

12.1 Bernoulli’s principle

In this section we will derive two theorems that make precise the principle discovered by Daniel Bernoulli in the 18th century that as the speed of a fluid increases, the pressure of the fluid decreases. This will later allow us to understand the origin of the lift force on an aerofoil, but of course has many other important practical applications.

To facilitate the derivation of Bernoulli’s theorems, we will make two observations that allow us to rewrite Euler’s equations in new ways.

  1. Because gravity is a conservative force, it can be written as minus the gradient of a potential: \(\underline{g}= -\underline{\nabla}\chi\). Hence, Eq. 11.11 can also be written as \[ \frac{D\underline{u}}{Dt} = - \underline{\nabla} \left(\frac{p}{\rho} + \chi\right). \tag{12.1}\] In the simple case where gravitational force is constant and in the negative \(z\) direction, \(\chi = g\,z\) so that \(\underline{g} = -g\,\underline{e}_z\).

  2. We will use the vector calculus identity \[ (\underline{u}\cdot\underline{\nabla})\underline{u} = (\underline{\nabla}\times\underline{u})\times\underline{u}+\underline{\nabla}\left(\frac{u^2}{2}\right), \tag{12.2}\] which holds for any differentiable vector field \(\underline{u}\) and thus in particular for the velocity field. Using this, together with the definition of the material derivative, in the left-hand side of Eq. 12.1 gives \[\begin{split} \frac{D\underline{u}}{Dt} &=\partial_t\underline{u}+(\underline{u}\cdot\underline{\nabla})\underline{u}\\ &=\partial_t\underline{u}+(\underline{\nabla}\times\underline{u})\times\underline{u}+\underline{\nabla}\left(\frac{u^2}{2}\right)\\ &=- \underline{\nabla} \left(\frac{p}{\rho} + \chi\right). \end{split} \tag{12.3}\] This we can rewrite as \[ \partial_t\underline{u}+(\underline{\nabla}\times\underline{u})\times\underline{u} = - \underline{\nabla} \left(\frac{p}{\rho} + \frac{u^2}{2} + \chi\right). \tag{12.4}\] In this form the Euler equation involves the curl of \(\underline{u}\) which is known as the vorticity \[ \underline{\omega}=\operatorname{curl} \underline{u} = \underline{\nabla}\times\underline{u}, \tag{12.5}\] which we will study in more detail in the next section. If we also introduce the function \[ H=\frac{p}{\rho} + \frac{u^2}{2} + \chi, \tag{12.6}\] sometimes known as Bernoulli’s integral, we can write Euler’s equation simply as \[ \partial_t\underline{u}+\underline{\omega}\times\underline{u} = - \underline{\nabla} H. \tag{12.7}\]

This simplifies even further if we consider a steady steady flow, so that \(\partial_t\underline{u}=0\). Then Eq. 12.7 simplifies to \[ \underline{\omega}\times\underline{u}=-\underline{\nabla} H. \tag{12.8}\] To get rid of the vector product in this equation, we take the dot product with \(\underline{u}\) on both sides: \[ \underline{u}\cdot(\underline{\omega}\times\underline{u})=-\underline{u}\cdot\underline{\nabla} H. \tag{12.9}\] On the left hand side the cross product \(\underline{\omega}\times\underline{u}\) is orthogonal to \(\underline{u}\) so that the dot product with \(\underline{u}\) is zero. So the left-hand side of the above equation is zero and we have \[ \underline{u}\cdot\underline{\nabla} H=0. \tag{12.10}\] In words this says that the derivative of \(H\) in the direction of the flow, i.e., along streamlines, is zero, which gives:

Theorem 12.1 (Bernoulli’s streamline theorem) If an ideal fluid is in steady flow then \[ H=\frac{p}{\rho} + \frac{u^2}{2} + \chi \tag{12.11}\] is constant along streamlines.

To keep \(H\) constant, higher velocity must correspond to lower pressure, and vice versa. So we notice Bernoulli’s principle. But notice that this is only true along each individual streamline. This theorem says nothing about how \(H\) varies between streamlines.

Next we restrict ourselves to flows where there is no vorticity, \(\underline{\omega} = 0\). Such flows are called irrotational flows. In this case Eq. 12.8 implies that \(\underline{\nabla} H=0\), which together with \(\partial_t H=0\) gives us

Theorem 12.2 (Bernoulli’s theorem for irrotational flow) If an ideal fluid is in steady irrotational flow then \[ H=\frac{p}{\rho} + \frac{u^2}{2} + \chi \tag{12.12}\] is constant.

This theorem determines the pressure in a steady, irrotational ideal fluid once the velocity is known.

We can use this theorem to explain the lift created by the airflow around an aerofoil: because the air flows faster along the top of the aerofoil than along the bottom, there is a lower pressure above the aerofoil than below. However this is not a complete explanation until we have answered two questions:

  1. Why is the flow irrotational?

  2. Why is the velocity higher along the top?

We’ll address the first question in the next section and the second question in the following lectures.

12.2 Vorticity

Definition 12.1 The vorticity \(\underline{\omega}\) of a flow with the velocity field \(\underline{u}\) is defined as the curl of \(\underline{u}\): \[ \underline{\omega}=\operatorname{curl} \underline{u} = \underline{\nabla}\times\underline{u}. \tag{12.13}\]

An example will make it easier to understand how to best think of vorticity:

Example 12.1 We consider the very simple flow \(\underline{u}=(\alpha y, 0, 0)\) for some constant \(\alpha>0\). This flow is parallel to the \(x\) axis, with the magnitude of the velocity increasing with increasing \(y\). The streamlines are simply straight horizontal lines parallel to the \(x\) axis. These are also the pathlines because this is a steady flow. So nothing flows in circles, so one might be tempted to guess that this is an irrotational flow. However the calculation shows that \[\begin{split} \underline{\omega}&= \underline{\nabla}\times\underline{u}=(\partial_y\underline{u}_z-\partial_z\underline{u}_y, \partial_z\underline{u}_x-\partial_x\underline{u}_z, \partial_x\underline{u}_y-\partial_y\underline{u}_x)\\ &=(0,0,-\partial_y\underline{u}_x)=(0,0,-\alpha)\neq \underline{0}. \end{split} \tag{12.14}\] So there is a non-zero vorticity – the flow is not irrotational. To see this intuitively, imagine placing an extended object into the flow. If you want to do this in your bathtub, I recommend gluing two matches together to form a cross and place this on the water surface. If you now create a flow, this object may start to rotate. It will do so in the flow in this example because the upper end of the object finds itself in an area of faster flow while the lower end is in an area of slower flow, so the object will rotate clockwise.

12.2.1 The evolution of vorticity

To derive an equation which governs the evolution of vorticity, we take the \(curl\) of Euler’s equation in the form given in Eq. 12.7. \[ \underline{\nabla}\times\partial_t\underline{u}+\underline{\nabla}\times(\underline{\omega}\times\underline{u})=-\underline{\nabla}\times\underline{\nabla} H. \tag{12.15}\] Using the fact that \(curl\) of \(grad\) is zero, we obtain \[ \partial_t\underline{\omega} + \underline{\nabla}\times(\underline{\omega}\times \underline{u}) = \underline{0}. \tag{12.16}\] We will now use the general vector calculus identity \[ \underline{\nabla}\times(\underline{a}\times\underline{b})=(\underline{\nabla}\cdot\underline{b})\underline{a}+(\underline{b}\cdot\underline{\nabla})\underline{a}-(\underline{\nabla}\cdot\underline{a})\underline{b}-(\underline{a}\cdot\underline{\nabla})\underline{b} \tag{12.17}\] which holds for any two differentiable vector fields \(\underline{a}\) and \(\underline{b}\). Applying this with \(\underline{a}=\underline{\omega}\) and \(\underline{b}=\underline{u}\) gives \[ \partial_t\underline{\omega} + (\underline{\nabla}\cdot\underline{u})\underline{\omega}+(\underline{u}\cdot\underline{\nabla})\underline{\omega}-(\underline{\nabla}\cdot\underline{\omega})\underline{u}-(\underline{\omega}\cdot\underline{\nabla})\underline{u}= \underline{0}. \tag{12.18}\] Using the incompressibility condition \(\underline{\nabla}\cdot\underline{u}=0\) and the fact that the \(div\) of a \(curl\) is zero and thus \(\underline{\nabla}\cdot\underline{\omega}=0\), the above simplifies to \[ \partial_t\underline{\omega} + (\underline{u}\cdot\underline{\nabla})\underline{\omega} = (\underline{\omega}\cdot\underline{\nabla})\underline{u}. \tag{12.19}\] Using the definition of the material derivative we obtain the

Vorticity equation

\[ \frac{D\underline{\omega}}{Dt} = (\underline{\omega}\cdot\underline{\nabla})\underline{u}. \tag{12.20}\]

Note that the pressure \(p\) does not appear in the vorticity equation.

This simplifies for a 2D flow \(\underline{u}=(u_x(x,y,t), u_y(x,y,t),0)\). In the calculation of the vorticity most terms vanish leaving only \[ \underline{\omega}=\underline{\nabla}\times\underline{u}=(0,0,\omega)~~~\text{ with } \omega = \partial_x u_y-\partial_y u_x. \tag{12.21}\] Then \[ (\underline{\omega}\cdot\underline{\nabla})\underline{u} = \omega \, \partial_z \underline{u}=\underline{0} \tag{12.22}\] because there is no \(z\) dependence in the 2D flow. As a result, the vorticity equation Eq. 12.20 reduces to \[ \frac{D\omega}{Dt}=0. \tag{12.23}\] Thus:

Theorem 12.3 (Conservation of vorticity) In a 2D flow, the vorticity of each fluid particle is conserved, i.e., the vorticity is constant along each pathline.

This now allows us to answer our first question on the way to understanding the lift on an aerofoil. If we assume that far away in front of the aerofoil the air is still and thus described by a flow with no vorticity, then the vorticity stays zero along the paths of all the air molecules also as the aerofoil passes. Thus the entire flow is irrotational and we can apply Bernoulli’s theorem for irrotational flows to argue that the aerofoil experiences a lift force. All we still need to establish is that the velocity is higher above the aerofoil than below. That is going to occupy us for the next three lectures.

Vorticity is a fundamental concept in fluid dynamics that plays a critical role in determining the behaviour of a fluid flow. It is closely related to the formation of vortices, the conservation of angular momentum, and the formation of turbulence. We’ll leave all of this for later modules.