Control Volume & Fluxes. Eulerian and Lagrangian Formulations

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Control Volume & Fluxes. Eulerian and Lagrangian Formulations
Aula Teórica 7 Control Volume & Fluxes. Eulerian and Lagrangian Formulations

Resultant force applied to a volume of fluid

General movement equation
This equation holds for a material system with a unit of mass. It is written in a Lagrangian Formulation, i.e. one has to follow that portion of fluid in order to describe its velocity. That is not easy for us since we use to be in a fix place observing the flow, i.e. we are in an Eulerian reference .

Lagrangian vs Eulerian Descriptions
Both describe time derivatives. Lagrangian approach describes the rate of change of a property in a material system, i.e. follows material as it moves. (It is the unique formulation to describe the movement of bodies) Eulerian describes the rate of change in one point of space. Lagrangian derivative is velocity independent. Eulerian derivative registers changes due to fluid movement. In stationary systems eulerian derivative is null meaning that local production balances transport.

Case of velocity In this flow:
Is there acceleration (rate of change of the velocity of a material system)? How do momentum flux change between entrance and exit? If the flow is stationary what is the local velocity change rate? How does momentum inside the control volume change in time? How does pressure vary along the flow? What is the relation between momentum production and the divergence of momentum fluxes? Can we say that lagrangian description is better then eulerian description or vice-versa?

Concentration Fecal Bacteria dies in the environment according to a first order decay, i.e. the number of bacteria that dies per unit of time is proportional to existing bacteria. This process is describe by the equation: This is a lagrangian formulation. This solution describes what is happening inside a water mass whether is moving or not. What happens in an Eulerian description? C C0 t

Eulerian description Let’s consider a river where the contaminated water would be moving as a patch (without diffusion) t4 t1 X2 X1 t3 t2 Concentration decays as the patch moves. Time series in points x1 and x2 would be: C X1 Maximum concentration difference between X1 and X2 depends on decay rate while difference in time to show up and showing time reduce as flow velocity increases. X2 t

Lagrangian vs Eulerian
Examples of videos illustrating the difference between eulerian and lagrangian descriptions (not always very clear)

Reynolds Theorem The rate of change of a property inside a material system is equal to the rate of change inside the control volume occupied by the fluid plus what is flowing in, minus what is flowing out.

Demonstration of Reynolds Theorem
Let’s consider a conduct and 3 portions fluid (systems), SYS 1, SYS3 and SYS 3 that are moving. Time = t SYS 3 CV SYS 2 SYS 1 Let’s consider a space control volume (not moving) that at time “t” is completed filled by the fluid SYS 2 Time = t+∆t SYS 3 Between time= t and time =(t+∆t) inside the control volume properties changed because some fluid flew in (SYS1) and other flew out (SYS2) and also if properties of those systems have changed. SYS 2 SYS 1 CV

Rates of change In a material system: Inside the control volume:
SYS 2 was coincident with CV at time t: At time t+∆t:

Computing the budget per unit of time and using the specific property (per unit of volume)

Identically for the material System

How much is flowing in and out?
The volumetric discharge is the integral of the volume flowing per unit of area integrated over the area. The Mass discharge is the integral of the mass flowing per unit of area integrated over the area. The Mass flowing per unit of area is the volume per unit of area times the mass per unit of volume. If material is flowing in, the internal product is negative and if is flowing out is positive. As a consequence:

And finally Or:

If the Volume is infinitesimal
Becomes: But: And thus: Dividing by the volume:

Derivada total The Total derivative is the rate of change in a material system (Lagrangian description) ; The Partial derivative is the rate of change in a control volume (eulerian description) ; The advective derivative account for the transport by the velocity.

Evolution Equation The rate of change inside the system is the (Production-Destruction) + (diffusion exchange). Designating Production – Destruction by (Sources – Sinks) and knowing that:

Differential Equation