Kinematics and Flow Visualization

Material acceleration in Eulerian field

In an Eulerian description the velocity is a field \vec{V}(x,y,z,t), so the acceleration of a fluid particle must be found from the material (substantial) derivative, not from the local rate of change alone. The particle acceleration is \vec{a} = \frac{D\vec{V}}{Dt} = \frac{\partial \vec{V}}{\partial t} + (\vec{V}\cdot\nabla)\vec{V}. The first term is the local acceleration, which vanishes when the field has no explicit time dependence. The second term is the convective acceleration, which arises because the particle moves into regions where the velocity is different. In Cartesian components, a_x = \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} + w\frac{\partial u}{\partial z}, with analogous expressions for a_y and a_z. The magnitude of the acceleration is |\vec{a}| = \sqrt{a_x^2 + a_y^2 + a_z^2}. For a field u = \sqrt{k_1}\,x, v = -\sqrt{k_2}\,y at a point where x=1 and y=-1, the components are a_x = k_1 x and a_y = k_2 y, so the magnitude follows directly from these two values.

In one-dimensional steady flow the local term is zero and the acceleration reduces to a_x = u\frac{du}{dx}. For example, if u = 2x, then a_x = 2x\cdot 2 = 4x. Such a one-dimensional problem can be solved by evaluating u and du/dx at the given position and multiplying them.

The component of acceleration along a streamline is the projection of \vec{a} onto the unit tangent vector, a_s = \vec{a}\cdot\hat{V} = \frac{\vec{a}\cdot\vec{V}}{|\vec{V}|}. In steady flow this equals a_s = V\frac{dV}{ds}. A negative a_s means the particle is decelerating along its path, and the magnitude asked for in a problem is the absolute value of this projection.

The same substantial-derivative idea applies to any scalar property T, such as temperature. Its total rate of change is \frac{DT}{Dt} = \frac{\partial T}{\partial t} + u\frac{\partial T}{\partial x} + v\frac{\partial T}{\partial y} + w\frac{\partial T}{\partial z}. All partial derivatives are evaluated at the given point and time, and the velocity components are used as the advecting velocity. For instance, if T = xy + t and u=1, v=0, then at (2,3) the value is \frac{DT}{Dt} = 1 + 1\cdot 3 + 0 = 4.

A velocity field that contains no time variable is steady, whatever constants it contains. Even so, a steady flow usually has nonzero acceleration because of the convective term. For the linear field u = \alpha - x, the x-acceleration is a_x = u\frac{\partial u}{\partial x} = x - \alpha, which depends on position and is therefore not constant. A field whose acceleration is constant must have convective terms that do not vary with position, which is not the case for this linear field.

Streamline definition and properties

A streamline is a curve that is tangent to the velocity vector at every point at a given instant. Its differential equation in two dimensions is \frac{dx}{u} = \frac{dy}{v}, and in three dimensions \frac{dx}{u} = \frac{dy}{v} = \frac{dz}{w}. Integrating this equation gives the family of streamlines, and the constants of integration label each streamline. Two points lie on the same streamline only if they share the same value of that constant. For example, for \vec{V} = y\,\hat{i} + x\,\hat{j}, the equation x\,dx = y\,dy gives x^2 - y^2 = C. The streamline through (1,0) has C=1, so the point (2,p) on it satisfies 4 - p^2 = 1, giving p = \sqrt{3}.