2.2: Dimensionless Numbers

2.2.1 The Reynolds Number Re

In fluid mechanics, the Reynolds number (Re) is a dimensionless number that gives a measure of the ratio of inertial (resistant to change or motion) forces to viscous (heavy and gluey) forces and consequently quantifies the relative importance of these two types of forces for given flow conditions. (The term inertial forces, which characterize how much a particular liquid resists any change in motion, are not to be confused with inertial forces defined in the classical way.)

The concept was introduced by George Gabriel Stokes in 1851 but the Reynolds number is named after Osborne Reynolds (1842–1912), who popularized its use in 1883. Reynolds numbers frequently arise when performing dimensional analysis of liquid dynamics problems, and as such can be used to determine dynamic similitude between different experimental cases.

They are also used to characterize different flow regimes, such as laminar or turbulent flow: laminar flow occurs at low Reynolds numbers, where viscous forces are dominant, and is characterized by smooth, constant liquid motion; turbulent flow occurs at high Reynolds numbers and is dominated by inertial forces, which tend to produce chaotic eddies, vortices and other flow instabilities.

The gradient of the velocity dv/dx is proportional to the velocity v divided by a characteristic length scale L. Similarly, the second derivative of the velocity d²v/dx² is proportional to the velocity v divided by the square of the characteristic length scale L.

\[\ \operatorname{Re}=\frac{\text { Inertial forces }}{\text { Viscous forces }}=\frac{\rho_{1} \cdot \mathrm{v} \cdot \frac{\mathrm{d} \mathrm{v}}{\mathrm{d} \mathrm{x}}}{\rho_{\mathrm{l}} \cdot v_{\mathrm{l}} \cdot \frac{\mathrm{d}^{2} \mathrm{v}}{\mathrm{d} \mathrm{x}^{2}}} \quad \text{with: } \frac{\mathrm{d} \mathrm{v}}{\mathrm{d} \mathrm{x}} \propto \frac{\mathrm{v}}{\mathrm{L}} \quad \frac{\mathrm{d}^{2} \mathrm{v}}{\mathrm{d} \mathrm{x}^{2}} \propto \frac{\mathrm{v}}{\mathrm{L}^{2}} \quad \Rightarrow \frac{\mathrm{v} \cdot \mathrm{L}}{v_{\mathrm{l}}}\]

The Reynolds number is a dimensionless number. High values of the parameter (on the order of 10 million) indicate that viscous forces are small and the flow is essentially inviscid. The Euler equations can then be used to model the flow. Low values of the parameter (on the order of 1 hundred) indicate that viscous forces must be considered.

2.2.2 The Froude Number Fr

The Froude number (Fr) is a dimensionless number defined as the ratio of a characteristic velocity to a gravitational wave velocity. It may equivalently be defined as the ratio of a body's inertia to gravitational forces. In fluid mechanics, the Froude number is used to determine the resistance of a partially submerged object moving through water, and permits the comparison of objects of different sizes. Named after William Froude (1810-1879), the Froude number is based on the speed–length ratio as defined by him.

\[\ \mathrm{F} \mathrm{r}=\frac{\text { Characteristic velocity }}{\text { Gravitational wave velocity }}=\frac{\mathrm{v}}{\sqrt{\mathrm{g} \cdot \mathrm{L}}}\]

Or the ratio between the inertial force and the gravitational force squared according to:

\[\ \widehat{\mathrm{F}} \mathrm{r}=\frac{\text { Inertial force }}{\text { Gravitational force }}=\frac{\rho_{\mathrm{l}} \cdot \mathrm{v} \cdot \frac{\mathrm{d} \mathrm{v}}{\mathrm{d x}}}{\rho_{\mathrm{l}} \cdot \mathrm{g}}=\frac{\mathrm{v}^{\mathrm{2}}}{\mathrm{g} \cdot \mathrm{L}}\]

The gradient of the velocity \(dv/dx\) is proportional to the velocity \(v\) divided by a length scale \(L\).

Or the ratio between the centripetal force on an object and the gravitational force, giving the square of the right hand term of equation (2.2-2):

\[\ \widehat{\mathrm{F} \mathrm{r}}=\frac{\text { Centripetal force }}{\text { Gravitational force }}=\frac{\mathrm{m} \cdot \mathrm{v}^{2} / \mathrm{L}}{\mathrm{m} \cdot \mathrm{g}}=\frac{\mathrm{v}^{\mathrm{2}}}{\mathrm{g} \cdot \mathrm{L}}\]

2.2.3 The Richardson Number Ri

The Richardson number Ri is named after Lewis Fry Richardson (1881-1953). It is the dimensionless number that expresses the ratio of the buoyancy term to the flow gradient term.

\[\ \mathrm {R} \mathrm {i}=\frac{\text { buoyancy term }}{\text { flow gradient term }}=\left(\frac{\mathrm {g} \cdot \mathrm {L} \cdot \mathrm {R}_{\mathrm {s} \mathrm {d}}}{\mathrm {v}^{\mathrm {2}}}\right)\]

The Richardson number, or one of several variants, is of practical importance in weather forecasting and in investigating density and turbidity currents in oceans, lakes and reservoirs.

2.2.4 The Archimedes Number Ar

The Archimedes number (Ar) (not to be confused with Archimedes constant, π), named after the ancient Greek scientist Archimedes is used to determine the motion of liquids due to density differences. It is a dimensionless number defined as the ratio of gravitational forces to viscous forces. When analyzing potentially mixed convection of a liquid, the Archimedes number parameterizes the relative strength of free and forced convection. When Ar >> 1 natural convection dominates, i.e. less dense bodies rise and denser bodies sink, and when Ar << 1 forced convection dominates.

\[\ \mathrm {A} \mathrm {r}=\frac{\text { Gravitational forces }}{\text { Viscous forces }}=\frac{\mathrm {g} \cdot \mathrm {L}^{3} \cdot \mathrm {R}_{\mathrm {s} \mathrm {d}}}{v_{\mathrm {l}}^{\mathrm {2}}}\]

The Archimedes number is related to both the Richardson number and the Reynolds number via:

\[\ \mathrm {A} \mathrm {r}=\mathrm {R} \mathrm {i} \cdot \mathrm {R} \mathrm {e}^{2}=\left(\frac{\mathrm {g} \cdot \mathrm {L} \cdot \mathrm {R}_{\mathrm {s} \mathrm {d}}}{\mathrm {v}^{\mathrm {2}}}\right) \cdot\left(\frac{\mathrm {v} \cdot \mathrm {L}}{v_{\mathrm {l}}}\right)^{\mathrm {2}}=\frac{\mathrm {g} \cdot \mathrm {L}^{\mathrm {3}} \cdot \mathrm {R}_{\mathrm {s} \mathrm {d}}}{v_{\mathrm {l}}^{\mathrm {2}}}\]

2.2.5 The Thủy Number Th or Collision Intensity Number

The new Thủy number (Th) is the cube root of the ratio of the viscous forces times the gravitational forces to the inertial forces squared. Thủy is Vietnamese for aquatic, water. The gradient of the velocity v is proportional to the velocity v divided by a length scale L. Since slurry transport is complex and inertial forces, viscous forces and gravitational forces play a role, this dimensionless number takes all of these forces into account in one dimensionless number.

\[\ \widehat{\mathrm{Th}}=\frac{\text { Viscous forces }}{\text { Inertial forces }} \cdot \frac{\text { Gravitational forces }}{\text { Inertial forces }}=\frac{1}{\operatorname{Re} \cdot \widehat{\mathrm{Fr}}}=\frac{\rho_{1} \cdot v_{1} \cdot \frac{\mathrm{d}^{2} \mathrm{v}}{\mathrm{dx}^{2}}}{\rho_{1} \cdot \mathrm{v} \cdot \frac{\mathrm{d} v}{\mathrm{d} \mathrm{x}}} \cdot \frac{\rho_{1} \cdot \mathrm{g}}{\rho_{1} \cdot \mathrm{v} \cdot \frac{\mathrm{d} \mathrm{v}}{\mathrm{d} \mathrm{x}}}=\frac{v_{1} \cdot \mathrm{g}}{\mathrm{v}^{3}}\]

So:

\[\ \mathrm{T} \mathrm{h}=\left(\frac{v_{\mathrm{l}} \cdot \mathrm{g}}{\mathrm{v}^{\mathrm{3}}}\right)^{\mathrm{1} / 3}\]

It is interesting that the length scale does not play a role anymore in this dimensionless number. The different terms compensate for the length scale. The value of this dimensionless parameter is, that the relative excess head losses are proportional with the Thủy number to a certain power. Also the Limit Deposit Velocity in heterogeneous transport has proportionality with this dimensionless number.

2.2.6 The Cát Number Ct or Collision Impact Number

A special particle Froude number will be introduced here. The Durand & Condolios (1952) particle Froude number Cát, Ct, which is Vietnamese for sand grains. This dimensionless number describes the contribution of the solids to the excess head losses.

\[\ \mathrm{C t}=\left(\frac{\mathrm{v}_{\mathrm{t}}}{\sqrt{\mathrm{g} \cdot \mathrm{d}}}\right)^{\mathrm{5} / 3}=\left(\frac{\mathrm{1}}{\sqrt{\mathrm{C}_{\mathrm{x}}}}\right)^{\mathrm{5} / 3}\]

The introduction of this particle Froude number is very convenient in many equations.

2.2.7 The Lắng Number La or Sedimentation Capability Number

Another new dimensionless number is introduced here. It is the Lắng number La. Lắng is Vietnamese for sediment and this number represents the capability of the slurry flow to form a bed, either fixed or sliding.

\[\ \mathrm{L a}=\frac{\mathrm{v}_{\mathrm{t}} \cdot\left(\mathrm{1}-\mathrm{C}_{\mathrm{v s}} / \mathrm{\kappa}_{\mathrm{C}}\right)^{\mathrm{\beta}}}{\mathrm{v}_{\mathrm{l s}}}\]

2.2.8 The Shields Parameter θ

The Shields parameter, named after Albert Frank Shields (1908-1974), also called the Shields criterion or Shields number, is a non-dimensional number used to calculate the initiation of motion of sediment in a fluid flow. It is a non dimensionalisation of a shear stress. By multiplying both the nominator and denominator by d², one can see that it is proportional to the ratio of fluid force on the particle to the submerged weight of the particle.

\[\ \theta=\frac{\mathrm{F}_{\text {shear }}}{\mathrm{F}_{\text {gravity }}} \propto \frac{\rho_{\mathrm{l}} \cdot \mathrm{u}_{*}^{2} \cdot \mathrm{d}^{2}}{\rho_{\mathrm{l}} \cdot \mathrm{R}_{\mathrm{s d}} \cdot \mathrm{g} \cdot \mathrm{d}^{3}}=\frac{\mathrm{u}_{*}^{2}}{\mathrm{R}_{\mathrm{s d}} \cdot \mathrm{g} \cdot \mathrm{d}}\]

The Shields parameter gives an indication of the erodibility of a sediment. If the Shields parameter is below some critical value there will not be erosion, if it’s above this critical value there will be erosion. The higher the Shields parameter, the bigger the erosion. The critical Shields parameter depends on the particle diameter, the kinematic viscosity and some other parameters. The boundary Reynolds number as used in Shields graphs.

\[\ \mathrm{R} \mathrm{e}_{*}=\frac{\mathrm{u}_{*} \cdot \mathrm{d}}{v_{\mathrm{l}}}\]

The roughness Reynolds number.

\[\ \mathrm{k}_{\mathrm{s}}^{+}=\frac{\mathrm{u}_{*} \cdot \mathrm{k}_{\mathrm{s}}}{v_{\mathrm{l}}}\]

The distance to the wall Reynolds number:

\[\ \mathrm{y}^{+}=\frac{\mathrm{u}_{*} \cdot \mathrm{y}}{v_{\mathrm{l}}}\]

2.2.9 The Bonneville Parameter D*

The original Shields graph is not convenient to use, because both axes contain the shear velocity u* and this is usually an unknown, this makes the graph an implicit graph. To make the graph explicit, the graph has to be transformed to another axis system. In literature often the dimensionless grain diameter D* is used, also called the Bonneville (1963) parameter:

\[\ \mathrm{D}_{*}=\mathrm{d} \cdot \sqrt[3]{\frac{\mathrm{R}_{\mathrm{sd}} \cdot \mathrm{g}}{v_{\mathrm{l}}^{2}}}\]

The relation between the Shields parameter and the Bonneville parameter is:

\[\ \mathrm{R} \mathrm{e}_{*}=\sqrt{\boldsymbol{\theta}} \cdot \mathrm{D}_{*}^{\mathrm{1 . 5}}\]

So the Bonneville parameter is a function of the Shields number and the boundary Reynolds number according to:

\[\ \mathrm{D}_{*}=\left(\frac{\mathrm{R} \mathrm{e}_{*}}{\sqrt{\mathrm{\theta}}}\right)^{2 / 3}\]

2.2.10 The Rouse Number P

The Rouse number, named after Hunter Rouse (1906-1996), is a non-dimensional number used to define a concentration profile of suspended sediment in sediment transport.

\[\ \mathrm{P}=\frac{\mathrm{v}_{\mathrm{t}}}{\mathrm{\kappa} \cdot \mathrm{u}_{*}} \quad or \quad \mathrm{P}=\frac{\mathrm{v}_{\mathrm{t}}}{\boldsymbol{\beta} \cdot \mathrm{\kappa} \cdot \mathrm{u}_{*}}\]

The factor β is sometimes included to correlate eddy viscosity to eddy diffusivity and is generally taken to be equal to 1 and is therefore usually ignored. The von Karman constant κ is about 0.4.

The value of the Rouse number is an indication of the type of sediment transport and the bed form.

Table 2.2-1: Some interpretations of the value of the Rouse number

P≥7.5	Little movement.
7.5≥P≥2.5	Bed load (grains rolling and hopping along the bed, bed forms like dunes) to suspension in the lower part.
2.5≥P≥0.8	Incipient suspension (grains spending less and less time in contact with the bed, bed forms increase in wavelength and decrease in amplitude). For P=2.5 there is suspension in the lower part of the channel or pipe, For P=0.8 the suspension reaches the surface.
0.8≥P	Suspension (grains spend very little time in contact with the bed, a plane bed). For P=0.1 the suspension is well developed, for P=0.01 the suspension is homogeneous.

2.2.11 The Stokes Number Stk

The Stokes number Stk, named after George Gabriel Stokes (1819-1903), is a dimensionless number corresponding to the behavior of particles suspended in a fluid flow. The Stokes number is defined as the ratio of the characteristic time of a particle to a characteristic time of the flow or of an obstacle:

\[\ \mathrm{S} \mathrm{t} \mathrm{k}=\frac{\mathrm{t}_{\mathrm{0}} \cdot \mathrm{u}_{\mathrm{0}}}{\mathrm{I}_{\mathrm{0}}}\]

Where t₀ is the relaxation time of the particle (the time constant in the exponential decay of the particle settling velocity due to drag), u₀ is the velocity of the fluid (liquid) of the flow well away from the particle and l₀ is a characteristic dimension of the flow (typically the pipe diameter). In the case of Stokes flow, which is when the particle Reynolds number is low enough for the drag coefficient to be inversely proportional to the Reynolds number itself, the relaxation time can be defined as:

\[\ \mathrm{t}_{\mathrm{0}}=\frac{\rho_{\mathrm{s}} \cdot \mathrm{d}^{\mathrm{2}}}{\mathrm{1 8} \cdot \rho_{\mathrm{l}} \cdot v_{\mathrm{l}}}\]

In experimental fluid dynamics, the Stokes number is a measure of flow fidelity in particle image velocimetry (PIV) experiments, where very small particles are entrained in turbulent flows and optically observed to determine the speed and direction of fluid movement. For acceptable tracing accuracy, the particle response time should be faster than the smallest time scale of the flow. Smaller Stokes numbers represent better tracing accuracy. For Stk »1, particles will detach from a flow especially where the flow decelerates abruptly. For Stk«0.1, tracing accuracy errors are below 1%. The Stokes number also gives a good indication for small particles being capable of forming a homogeneous mixture with the liquid flow. Assuming, in the case of pipe flow, the line speed as the characteristic velocity u₀ and half the pipe diameter as the characteristic dimension l₀, this gives:

\[\ \mathrm{S} \mathrm{t} \mathrm{k}=\frac{\rho_{\mathrm{s}} \cdot \mathrm{d}^{2}}{\mathrm{1 8} \cdot \rho_{\mathrm{l}} \cdot v_{\mathrm{l}}} \cdot \frac{\mathrm{2} \cdot \mathrm{v}_{\mathrm{l s}}}{\mathrm{D}_{\mathrm{p}}} \quad \text{or} \quad \mathrm{d}=\sqrt{\frac{\mathrm{S t k} \cdot \mathrm{9} \cdot \rho_{\mathrm{l}} \cdot v_{\mathrm{l}} \cdot \mathrm{D}_{\mathrm{p}}}{\rho_{\mathrm{s}} \cdot \mathrm{v}_{\mathrm{l s}}}}\]

2.2.12 The Bagnold Number Ba

The Bagnold number (Ba) is the ratio of grain collision stresses to viscous fluid stresses in a granular flow with interstitial Newtonian fluid, first identified by Ralph Alger Bagnold. The Bagnold number is defined by:

\[\ \begin{aligned}\mathrm{B a}=\frac{\rho_{\mathrm{s}} \cdot \mathrm{d}^{2} \cdot \lambda^{1 / 2} \cdot \dot{\gamma}}{\mu_{\mathrm{l}}}=\frac{\rho_{\mathrm{s}} \cdot \mathrm{d}^{2} \cdot \lambda^{1 / 2} \cdot \dot{\gamma}}{\rho_{\mathrm{l}} \cdot v_{\mathrm{l}}} \quad \text {with : }\dot{\gamma}=\frac{\mathrm{d} \mathrm{v}}{\mathrm{d} \mathrm{r}}
\\ \text{With :} \lambda=\frac{1}{\left(\left(\frac{\mathrm{C}_{\mathrm{v b}}}{\mathrm{C}_{\mathrm{v s}}}\right)^{1 / 3}-\mathrm{1}\right)}=\frac{\mathrm{1}}{\left(\left(\frac{\mathrm{1}}{\mathrm{C}_{\mathrm{v r}}}\right)^{1 / 3}-\mathrm{1}\right)}=\frac{\mathrm{C}_{\mathrm{v r}}^{\mathrm{1 / 3}}}{\mathrm{1}-\mathrm{C}_{\mathrm{v r}}^{\mathrm{1 / 3}}}
\\ \text{Boundary layer: } \dot{\gamma}=\frac{\mathrm{u}_{*}^{2}}{v_{1}} \Rightarrow \mathrm{B a}=\frac{\rho_{\mathrm{s}} \cdot \mathrm{d}^{2} \cdot \lambda^{1 / 2} \cdot \mathrm{u}_{*}^{2}}{\rho_{\mathrm{l}} \cdot v_{\mathrm{l}}^{2}}\end{aligned}\tag{2.2.23}\]

Where C_vs is the solids fraction and C_vb is the maximum possible concentration, the bed concentration. In flows with small Bagnold numbers (Ba < 40), viscous fluid stresses dominate grain collision stresses, and the flow is said to be in the 'macro-viscous' regime. Grain collision stresses dominate at large Bagnold number (Ba > 450), which is known as the 'grain-inertia' regime. A transitional regime falls between these two values.