Relationships between Root Mean Square Velocities, Average Velocities, and Coefficients of Heterogeneity

Abstract

As we saw in chapter 6:

• the average vertical velocity is the time average of the instantaneous velocities v:

  $${V_M} = \frac{1}{T}\int_o^T {v(t)dt} $$

  or

  $${V_M} = \frac{1}{T}\sum\limits_{k = 1}^n {{v_k}{t_k}} $$

• the root mean square velocity VRMS is the square root of the time average of the squares of the instantaneous velocities:

  $$V_{RMS}^2 = \frac{1}{T}\int_o^T {{v^2}(t)dt} $$

  or

  $$V_{RMS}^2 = \frac{1}{T}\sum\limits_{k = 1}^n {v_k^2(t){t_k}} $$

It is important to note that for a given depth, the values of V_M and V_RMS are independent by definition of the distribution function of the velocities v_k.