Velocity Component Along Horizontal X Axis Calculator

Formula:

\[ u_x = V_s \times e^{\pi \times z / D_F} \times \cos(45 + (\pi \times z / D_F)) \]

Velocity at the Surface (V_s):

m/s

Vertical Coordinate (z):

Depth of Frictional Influence (D_F):

Velocity Component along Horizontal x Axis (u_x):

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1. What is Velocity Component along Horizontal x Axis?

The Velocity Component along a Horizontal x Axis represents the speed in the direction parallel to the x-axis in a two-dimensional system, particularly in fluid dynamics and oceanography where frictional influences are considered.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ u_x = V_s \times e^{\pi \times z / D_F} \times \cos(45 + (\pi \times z / D_F)) \]

Where:

\( u_x \) — Velocity Component along Horizontal x Axis (m/s)
\( V_s \) — Velocity at the Surface (m/s)
\( z \) — Vertical Coordinate (m)
\( D_F \) — Depth of Frictional Influence (m)
\( e \) — Napier's constant (≈2.71828)
\( \pi \) — Archimedes' constant (≈3.14159)

Explanation: This formula calculates the horizontal velocity component considering exponential decay and phase shift due to frictional effects with depth.

3. Importance of Velocity Component Calculation

Details: Accurate calculation of velocity components is crucial for understanding fluid flow patterns, sediment transport, and coastal engineering applications where frictional effects play a significant role.

4. Using the Calculator

Tips: Enter surface velocity in m/s, vertical coordinate in meters, and depth of frictional influence in meters. All values must be valid (V_s > 0, D_F > 0).

5. Frequently Asked Questions (FAQ)

Q1: What is the physical significance of this formula?
A: This formula describes how horizontal velocity components vary with depth in a fluid subject to frictional forces, commonly used in Ekman layer theory.

Q2: What are typical values for Depth of Frictional Influence?
A: D_F typically ranges from 10-200 meters in oceanographic applications, depending on water viscosity and Coriolis effects.

Q3: Why does the formula include a 45-degree phase shift?
A: The 45-degree phase shift accounts for the Coriolis effect that causes velocity vectors to rotate with depth in rotating systems.

Q4: Can this formula be used for atmospheric applications?
A: Yes, with appropriate scaling, this formulation can be applied to atmospheric boundary layers where similar frictional processes occur.

Q5: What are the limitations of this model?
A: This model assumes constant eddy viscosity and neglects stratification effects, making it most accurate for well-mixed homogeneous fluids.