# Streamline

Streamlines made visible in the wind tunnel on a model of the Schlörwagens
Streamlines around a car
Streamline around a wing profile

The streamline is a term from fluid mechanics . Streamlines are geometrical tools for the descriptive description of a flow (a directed movement of particles or continuously moving fluids ). In steady flow , a streamline can be thought of as the path of a small, light particle in the fluid that it would take with the fluid.

Streamlines are the curves in the velocity field of a flow , the tangent direction of which coincides with the directions of the velocity vectors, that is, they run tangential to the velocity field at every point. They give a clear impression of the current flow field and indicate problematic flow areas (e.g. flow separation ).

All streamlines of a stream together form the stream tube .

The streamlines, together with railway lines , stroking lines and timelines, are part of the “Characteristic Lines” visualization concept .

With a steady flow the streamlines coincide with the particle trajectories. In the case of unsteady flow, on the other hand, this is not the case, since the streamlines show an image of the currently existing velocity directions, whereas the particle trajectories represent the velocity directions taken by a particle in the course of time.

With steady currents, streamlines can be determined experimentally in the wind tunnel , e.g. B. in a car flow, make visible. Usually, however, you can see railway lines or streak lines in the wind tunnel .

## properties

• Streamlines cannot have a kink and also cannot intersect, since two different flow velocities cannot prevail at one point at the same time.
• A contraction (moving closer together) of the streamlines means an acceleration of the flow in the subsonic, but a deceleration in the supersonic.
• Diverging streamlines show a deceleration of the flow in the subsonic, but an acceleration in the supersonic.
• With curved streamlines, the pressure increases in the centrifugal direction.
• With straight, parallel streamlines, there is no change in pressure across the streamline.
• The lines of constant potential run orthogonally to streamlines.

## calculation

Be with a three-dimensional flow field. ${\ displaystyle {\ underline {u}} ({\ underline {x}}, t) = {\ begin {pmatrix} u \\ v \\ w \ end {pmatrix}}}$${\ displaystyle {\ underline {x}} = {\ begin {pmatrix} x \\ y \\ z \ end {pmatrix}}}$

The streamlines are curves that are tangential to the current velocity field at every point. So it applies

${\ displaystyle \ mathrm {d} {\ underline {x}} \ times {\ underline {u}} \ equiv 0}$ or.

in parameter-free representation

${\ displaystyle {\ frac {\ mathrm {d} x} {u}} = {\ frac {\ mathrm {d} y} {v}} = {\ frac {\ mathrm {d} z} {w}} }$

## Mathematical derivation of the streamline differential equation

Corresponds to the general parameterized representation of a current line as a curve , here is an arbitrary starting point on the power line, the curve and the array parameters. If an explicit streamline is considered at a certain point in time , this is used and the curve is thus completely described by the parameter . ${\ displaystyle {\ underline {x}} ({\ underline {x_ {0}}}, s, t)}$${\ displaystyle {\ underline {x_ {0}}}}$${\ displaystyle {s}}$${\ displaystyle {t}}$${\ displaystyle {T}}$${\ displaystyle {s}}$

The unit tangent vector of the curve corresponds to the moment? . ${\ displaystyle {\ underline {\ tau}}}$${\ displaystyle {\ underline {\ tau}} = {\ frac {\ mathrm {d} {\ underline {x}}} {| \ mathrm {d} {\ underline {x}} |}} = {\ frac {\ mathrm {d} {\ underline {x}}} {\ mathrm {d} s}}}$

If one also considers the speed on all points of the streamline at the selected point in time , one recognizes that the normalized vector is the speed . ${\ displaystyle {\ underline {u}} ({\ underline {x}}, t)}$${\ displaystyle {T}}$${\ displaystyle {\ frac {\ mathrm {\ underline {u}}} {| {\ underline {u}} |}} = {\ underline {\ tau}}}$

If one now equates these two expressions for the tangent unit vector, it follows: ${\ displaystyle {\ underline {\ tau}} = {\ frac {\ underline {u}} {| {\ underline {u}} |}} = {\ frac {\ mathrm {d} {\ underline {x} }} {| \ mathrm {d} {\ underline {x}} |}} = {\ frac {\ mathrm {d} {\ underline {x}}} {\ mathrm {d} s}}}$

and thus: ${\ displaystyle {\ frac {\ underline {u}} {| {\ underline {u}} |}} = {\ frac {\ mathrm {d} {\ underline {x}}} {\ mathrm {d} s }}}$

or in tensor notation: . ${\ displaystyle {\ frac {\ mathrm {d} {x_ {i}}} {\ mathrm {d} s}} = {\ frac {u_ {i}} {| {\ sqrt {u_ {k} u_ { k}}} |}}}$

This vector equation leads to three scalar equations:

1. ${\ displaystyle {\ frac {\ mathrm {d} x_ {1}} {\ mathrm {d} s}} = {\ frac {u_ {1}} {\ sqrt {u_ {1} ^ {2} + u_ {2} ^ {2} + u_ {3} ^ {2}}}}}$
2. ${\ displaystyle {\ frac {\ mathrm {d} x_ {2}} {\ mathrm {d} s}} = {\ frac {u_ {2}} {\ sqrt {u_ {1} ^ {2} + u_ {2} ^ {2} + u_ {3} ^ {2}}}}}$
3. ${\ displaystyle {\ frac {\ mathrm {d} x_ {3}} {\ mathrm {d} s}} = {\ frac {u_ {3}} {\ sqrt {u_ {1} ^ {2} + u_ {2} ^ {2} + u_ {3} ^ {2}}}}}$

If you now divide the equations by one another, you get the form:

(1) / (2): ${\ displaystyle {\ frac {\ mathrm {d} x_ {1}} {\ mathrm {d} x_ {2}}} = {\ frac {u_ {1}} {u_ {2}}}}$

(2) / (3): ${\ displaystyle {\ frac {\ mathrm {d} x_ {2}} {\ mathrm {d} x_ {3}}} = {\ frac {u_ {2}} {u_ {3}}}}$

(3) / (1): ${\ displaystyle {\ frac {\ mathrm {d} x_ {3}} {\ mathrm {d} x_ {1}}} = {\ frac {u_ {3}} {u_ {1}}}}$

The shape given in the section on calculation can be found by simply reshaping .

## Equation of motion perpendicular to the streamline

To create a curved streamline in a steady flow, a corresponding centripetal force must be present. This is caused by a pressure gradient perpendicular to the streamline (i.e. in the radial direction). The amount of this radial pressure gradient depends on the flow velocity , the density of the fluid and the radius of curvature of the streamline: ${\ displaystyle c}$${\ displaystyle \ rho}$${\ displaystyle r_ {K}}$

${\ displaystyle {\ frac {\ partial p} {\ partial r}} = {\ frac {\ rho \ cdot c ^ {2}} {r_ {K}}}}$

The pressure perpendicular to a curved streamline consequently increases in the radial direction.