For flow in circular tubes, transition to turbulence usually...

For flow in circular tubes, transition to turbulence usually Occurs around $R e \approx 2300 .$ Investigate the circumstances under which the flows of (a) standard air and (b) water at $15^{\circ} \mathrm{C}$ become turbulent. On log-log graphs, plot: the average velocity, the volume flow rate, and the mass flow rate, at which turbulence first occurs, as functions of tube diameter.

Key Concepts

Dimensional Analysis and Scaling

This is a method used to derive relationships among physical quantities by considering their dimensions. In the context of flow through tubes, it helps in understanding how flow characteristics such as velocity and flow rates scale with changes in the tube diameter.

Flow Rate Metrics

These include measures such as average velocity, volumetric flow rate, and mass flow rate. Average velocity is the mean speed of the fluid, the volumetric flow rate quantifies the volume of fluid passing through a cross-section per unit time, and the mass flow rate measures the mass of fluid passing through per unit time. Each metric provides different insights into the dynamics of fluid movement in conduits.

Log-Log Graphs

These graphs plot both the horizontal and vertical axes on logarithmic scales. They are particularly useful for visualizing power-law relationships and multiplicative scaling effects over multiple orders of magnitude, thereby simplifying the analysis of how flow properties vary with tube diameter.

Reynolds Number

A fundamental dimensionless parameter in fluid mechanics defined as Re = (density × velocity × characteristic length) / viscosity. It determines the relative importance of inertial versus viscous forces in a flow, which is critical for predicting whether the fluid flow will be laminar or turbulent.

Transition to Turbulence

This concept describes the change in flow behavior from smooth, orderly laminar flow to chaotic, eddy-filled turbulent flow as the Reynolds number exceeds a critical value. Understanding this transition is key to predicting and controlling flow behavior in various engineering applications.

Transcript

00:01 We have problem number four which is from the fluid dynamics of the physics okay so we have been provided with the renalz number which is r e approximately it is two thousand three hundred and it is given that the turbulence usually occurs around this so we can say that we have critical renalts number renounce number critically equal to 2 ,300.

00:31 This is a renalt number which decides whether the flow is turbulent or a streamline.

00:37 So we have to investigate the circumstance that the flows of a standard air and water becomes turbulent.

00:47 Okay, so we have to basically find the mass flow rate, mass flow rate of air and mass flow rate of water for this particular rennals number.

01:02 Mass flow rate of water for this particular renounce number.

01:05 Now we know that the formula for renounce number is are equal to v into d by new.

01:16 Okay, where this v is the velocity, d is the diameter of the pipe and new is the fluid viscosity okay new is the kinematic viscosity that is meter square per second okay so now we have to use this or this relationship to get all these things done okay now we have okay so now we can write it as in form of critical r e critical equal to vd by new.

02:07 So from here we can find out v equal to that is velocity equal to r e k re critical into new divided by d that is diameter but diameter is not being given.

02:28 Okay now we have to get certain things because the kinematic viscosity of air is fixed and kinematic viscosity of everything is fixed so let us get some let us write some parameters or some fixed things that is density of air we can say this is 1 .23 kg per meter cube this we will be using afterwards and then a kinematic viscosity of air, we can say that kinematic viscosity of air, air equal to 1 .45 into 10 to the power minus 5 meter square per second, and density of water.

03:19 And kinematic viscosity of water, we will be writing afterward...

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