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Mixing-Length Theory

It is method of striving to explain momentum change by turbulence Reynolds stresses inside a fluid boundary layer through an eddy viscosity. This idea was conceived by Ludwig Prandtl during the early 20th century. And after that he himself had reservations about the theory narrating fields ever since comprising atmospheric science, oceanography and stellar structure. The mixing-length theory is conceptually corresponding to the idea of mean free path in thermodynamics. Hence a fluid parcel will conserve its properties for a characteristic length before mixing with the surrounding fluid. Prantdl explained mixing length,

“may be considered as the diameter of the masses of fluid moving as a whole in each individual case; or again, as the distance traversed by a mass of this type before it becomes blended in with neighboring masses”.

The mechanism ad the mode of operations

According to the theory temperature is conserved for a certain distance as a parcel moves across a temperature gradient. The fluctuation in temperature which the parcel experienced all through the process is the temperature. So the temperature can be seen as the deviation from his surrounding environment after it has moved over his mixing length. And to understand the theory people must be able express quantities as the sums of their slowly varying components and fluctuating components. For instance

T= T1 + T2

Here T1 is the slowly varying element and T2 is the fluctuating element.

Thus it offers an order of magnitude solution to convective motion. It is a primary theory of turbulence which accounts for mirages and leads to the Kolmogrov spectrum. Consider a length scale and isolation of length and impose a heat flux. Then consider a typical velocity and the drive for a bubble’s motion is buoyancy due to the density variations.

The significance of the theory

The physical model, notions and unpredictability with actuality involved in mixing-length theory as applied to stellar convection are assessed. Consideration and emphasis are given to the opinions back mixing length theories, equations of motion, local mixing-length formalisms for a stationary envelope, the assumed structure of the convective flow, and the dynamics of convective eddies.

A particular mixing-length theory sample is thought about wherein the flow is represented by a conglomerate of cells or eddies which create and develop and gradually break up. The development of convective eddies is verified along with the eddy convection rate, initial situations, an eddy-annihilation hypothesis and turbulent fluxes. The choice of a mixing length and the calibration of the heat-flux formula are discussed and here Reynolds stress is analyzed and transport of heat and momentum by small-scale turbulence is included into the dynamics.

Questions:

• What is mixing-length theory?
• State the significance of the mixing-length theory.
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