# Flow velocity

The flow velocity , also flow velocity or flow velocity , is the velocity in a flow , a directed movement of particles or continuous bodies ( fluids ).

A distinction is made between the flow velocities of the individual particles and the mean flow velocity over a line, area or volume element or time interval.

The flow speed of water is the average speed at which the water moves downstream and is on the order of 1 meter per second. In contrast, that of groundwater is millimeters per second or centimeters per day, ie a few orders of magnitude lower.

## definition

The flow velocity is the change in location of the individual point (location) along its path . ${\ displaystyle {\ vec {x}} = (x, y, z)}$

${\ displaystyle \ omega \, {\ text {or}} \, v \, {\ text {or}} \, c = | {\ vec {v}} | = {\ bigl |} {\ dot {\ vec {x}}} {\ bigr |}}$
${\ displaystyle {\ vec {v}} = (v_ {x}, v_ {y}, v_ {z})}$: Flow
the point is the time derivative in physical notation

The flow velocity vectors carry one time line over into the next.

Average flow velocities can be determined using a streamline , the flow cross-section or the flow rate ( volume flow element , mass flow ).

## Flow velocity in the potential field

In the potential field , the flow velocity follows the equation of motion

${\ displaystyle {\ dot {\ vec {v}}} = - {\ vec {\ nabla}} \ Phi}$
${\ displaystyle \ nabla}$: Nabla operator
${\ displaystyle \ Phi}$: Potential

The specific energy equation applies :

${\ displaystyle {\ frac {v ^ {2}} {2}} + \ Phi = {\ text {const.}}}$

## Flow velocity in the Newtonian fluid

The flow velocity in a field of a flow in a Newtonian fluid is calculated from the Navier-Stokes equations , in their general formulation:

${\ displaystyle \ rho {\ frac {\ partial \ mathbf {v}} {\ partial t}} + \ rho (\ mathbf {v} \ cdot \ nabla) \ mathbf {v} = - \ nabla p + \ eta \ Delta \ mathbf {v} + (\ lambda + \ eta) \ nabla (\ nabla \ cdot \ mathbf {v}) + \ mathbf {f}.}$
${\ displaystyle \ rho}$: Density of the fluid
${\ displaystyle t}$: Time
${\ displaystyle p}$: Pressure
${\ displaystyle \ lambda}$: 1. Lamé constant
${\ displaystyle \ eta}$: 2. Lamé constant, dynamic viscosity
${\ displaystyle {\ frac {\ partial} {\ partial t}}}$: Partial derivative with respect to time
${\ displaystyle \ nabla}$: Nabla operator

For this basic equation of fluid mechanics , a system of non-linear partial differential equations of the 2nd order, there are numerous simplifications, special cases and numerical approaches.

Non-Newtonian fluids such as blood, glycerine or dough, which show a non-proportional, erratic flow behavior (see rheology ), behave differently .

## Application formulas

Examples of formulas in more specific application areas are:

• The Bernoulli equation for smooth liquids and gases:
${\ displaystyle h + {\ frac {\ omega ^ {2}} {2g}} + {\ frac {p} {\ gamma}} = {\ text {const}}.}$
• Bernoulli's general law for frictionless, eddy-free , stationary flows
${\ displaystyle {\ frac {\ omega ^ {2}} {2}} + {\ frac {p} {\ rho}} = {\ text {const}}.}$
• The Bernoulli law at constant density
${\ displaystyle \ rho {\ frac {v ^ {2}} {2}} + p = {\ text {const.}}}$
• Bernoulli equation with acceleration term for non-stationary flow:
${\ displaystyle h + {\ frac {\ omega ^ {2}} {2g}} + {\ frac {p} {\ gamma}} + {\ frac {1} {g}} \ int {\ frac {\ partial \ omega} {\ partial t}} = {\ text {const}}.}$
• Bernoulli equation of relative motion
${\ displaystyle {\ frac {\ omega ^ {2}} {2g}} + {\ frac {p} {\ gamma}} - {\ frac {u ^ {2}} {2g}} = {\ text { const}}.}$
${\ displaystyle u}$: Circumferential speed - application for blade rings
• Bernoulli equation at higher speeds (about> 150 m / s):
${\ displaystyle {\ frac {\ omega ^ {2}} {2g}} + \ int {\ frac {\ mathrm {d} p} {\ gamma}} = {\ text {const}}.}$

Typical formulas for measurement methods

• Dynamic pressure in the open air flow:
• ${\ displaystyle \ omega = {\ sqrt {\ frac {2q} {\ gamma}}}}$
${\ displaystyle q}$: Back pressure
• Lateral discharge from an open vessel:
• ${\ displaystyle \ omega = {\ sqrt {2 {\ frac {\ Delta p} {\ gamma}}}}}$

Medium speeds

• mean flow velocity with a non-constant cross-section
${\ displaystyle {\ bar {\ omega}} = {\ frac {1} {A}} \ cdot \ int _ {A} \ omega ({\ vec {x}}) \ cdot \ mathrm {d} A}$
Velocity at a point on the cross-section as a function of the location , with the direction of flow , so that .${\ displaystyle f (x, y)}$${\ displaystyle z}$${\ displaystyle A = A (z)}$

The flow velocity has an influence on the velocity-dependent parameters Reynolds number and Froude number .

## Measurement methods

The determination of the flow rate can be done with different techniques. Flow velocity in free air flow:

Flow velocity of currents in pipelines :

General measurement methods: