Kerosene is pumped through a smooth tube with inside diameter D=30 mm at close to the critical R

step 1 Let us write the shear stress distribution in a fully developed pi flow. So for fully developed

Kerosene is pumped through a smooth tube with inside...

Kerosene is pumped through a smooth tube with inside diameter $D=30 \mathrm{mm}$ at close to the critical Reynolds number. The flow is unstable and fluctuates hetween laminar and turbulent states, causing the pressure gradient to intermittently change from approximately $-4.5 \mathrm{kPa} / \mathrm{m}$ to $-11 \mathrm{kPa} / \mathrm{m} .$ Which pressure gradient corresponds to laminar. and which to turbulent, flow? For each flow, compute the shear stress at the tube wall, and sketch the shear stress distributions.

Key Concepts

Reynolds Number

The Reynolds number is a dimensionless quantity that characterizes the relative importance of inertial forces to viscous forces in a fluid flow. It helps predict the flow regime, with lower values typically indicating laminar (smooth and orderly) flow and higher values signaling turbulent (chaotic and mixing) flow. In many applications, there is a critical Reynolds number at which the flow transitions from laminar to turbulent, making it essential for understanding and predicting flow behaviour.

Laminar Flow

Laminar flow is characterized by smooth, orderly layers of fluid that slide past one another with little mixing. In laminar pipe flow, the velocity profile is parabolic, and the viscous shear stress distribution is linear from the centerline to the tube wall. Under a given pressure gradient, this results in lower wall shear stresses compared to turbulent flow. The lower pressure drop in the problem corresponds to the laminar state where frictional losses are lower.

Turbulent Flow

Turbulent flow is marked by irregular fluctuations, eddies, and chaotic mixing, which enhance momentum transfer and result in higher energy losses. In turbulent flow, the velocity profile is flatter in the core of the pipe with steeper gradients near the wall, leading to higher wall shear stresses. The higher pressure drop in the example is indicative of turbulent flow due to increased frictional forces and mixing.

Pressure Gradient

The pressure gradient in a fluid flow is the rate at which pressure changes along the length of the flow path. It acts as the driving force for the fluid movement and is directly related to frictional effects in the flow system. In pipe flow, a larger (in magnitude) negative pressure gradient indicates higher frictional losses and is typically associated with turbulent flow, while a smaller pressure gradient corresponds to laminar flow conditions.

Wall Shear Stress

Wall shear stress is the tangential force per unit area exerted by the flowing fluid on the boundary (such as the pipe wall). It is linked directly to the pressure gradient and flow regime; for both laminar and turbulent flows, it can be computed by the relation ?w = (D/4)|dp/dx| under fully developed conditions. Its value is lower for laminar flow and higher for turbulent flow because turbulence increases the effective momentum transfer near the wall.

Shear Stress Distribution

Shear stress distribution refers to how the tangential stress varies across the cross-section of the pipe. In laminar flow, the distribution is linear, being zero at the center of the pipe and reaching a maximum at the wall. For turbulent flow, the distribution is more complex due to enhanced mixing; while the maximum shear stress still occurs at the wall, the profile in the core is less predictable and typically flatter owing to the turbulent mixing effects.

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