Izbash formula

Last updated March 23, 2024

The Izbash formula is a mathematical expression used to calculate the stability of armourstone in flowing water environments.

For the assessment of granular material stability in a current, the Shields formula and the Izbash formula are commonly employed. The former is more appropriate for fine-grained materials like sand and gravel, whereas the Izbash formula is tailored for larger stone sizes. The Izbash formula was devised by Sergei Vladimirovich Izbash. Its general expression is as follows:^[1]

{\frac {u_{c}}{\sqrt {\Delta gd}}}=1.7

or alternatively

\Delta d=0.7{\frac {u^{2}}{2g}}

Here, the variables represent:

u_c = flow velocity in proximity to the stone

Δ = relative density of the stone, calculated as (ρ_s - ρ_w)/ρ_w where ρ_s denotes the stone's density and ρ_w is the water's density

g = gravitational acceleration

d = diameter of the stone

The coefficient 1.7 is an experimental constant determined by Izbash, encapsulating effects such as friction, inertia, and the turbulence of the current. Hence, the application of this coefficient is limited to conditions where turbulence is predominantly induced by the roughness of the construction materials in water. Adjustments are necessary when these conditions do not apply.

Derivation of the Izbash Formula

Forces acting on a stone in flowing water Izbash-derivation.jpg — Forces acting on a stone in flowing water

The derivation of the formula begins by considering the forces at play on a stone in a flowing current. These are grouped into active forces that tend to dislodge the stone, and passive forces that resist this movement:

Active Forces:
- Lift Force (F_L): Arises due to the flow of water around the stone, creating a pressure difference.
- Friction Force (F_S): Results from the contact between the stone and the riverbed.
- Drag Force (F_D): Generated by the flow of water against the stone's surface.
Passive Forces:
- The Stone's Weight (W): The downward force due to gravity.
- Resistance Force (F_F): The opposition offered by the bed's surface or other stones.^[1]

Each active force can be quantified in terms of the water's density (ρ_w), the flow velocity (u), and respective coefficients and areas of influence (C_D, C_F, C_L, A_D, A_S, A_L). The three active forces and two passive forces described above are considered. Analysing the moment equilibrium around point A results in F_F being disregarded due to its zero arm length. The active forces can then be detailed as:

{\begin{matrix}F_{D}&={\frac {1}{2}}C_{D}\rho _{w}u^{2}A_{D}\\F_{S}&={\frac {1}{2}}C_{F}\rho _{w}u^{2}A_{S}\\F_{L}&={\frac {1}{2}}C_{L}\rho _{w}u^{2}A_{L}\end{matrix}}\quad {\Biggr ]}\quad F\sim \rho _{w}u^{2}d^{2}

The total active force is proportional to the square of the flow velocity and the stone's diameter, represented as ρ_wu²d². The resisting passive force is proportional to the stone's submerged weight, which involves the gravitational constant (g), the stone's volume (proportional to d³), and the difference in density between the stone and the water (ρ_s - ρ_w), represented by Δ.

Balancing the active forces against the passive ones yields the critical flow velocity equation:

u_{c}^{2}=K\Delta gd

K is an empirical coefficient calibrated through experimental observations, and has been found to be around 1.7.

The formula therefore provides a critical velocity estimate: the threshold at which the forces acting on a stone due to flow surpass the stone's resistance to movement.^[2]

Calculation Example

Consider determining the requisite stone size to protect the base of a channel with a depth of 1 m and an average flow rate of 2 m/s.

The stone diameter necessary for protection can be estimated by reconfiguring the formula and inserting the relevant data. The Izbash formula necessitates the use of the velocity "near the stone," which is ambiguous. Practically, a velocity approximately equivalent to the stone's diameter above the protective layer is assumed. This translates to about 85% of the channel's average flow velocity when employing a standard logarithmic flow profile, resulting in a stone diameter of approximately 6.3 cm (comparable to the 6.5 cm predicted by the Shields formula).

Limitations

The application of the formula necessitates the measurement of velocity in proximity to the stone, a task that can be challenging, particularly in fine-grained soils and at significant water depths. Under such conditions, the Shields formula is often considered a more suitable alternative.^[3]

Modification by Pilarczyk

Recognising the prevalent usage of the coefficient 0.7, Krystian Pilarczyk refined the formula in 1985 for enhanced specificity.^[2] The revised equation is expressed as:

\Delta d=0.035{\frac {\Phi }{\Psi }}{\frac {K_{t}K_{h}}{K_{s}}}{\frac {u^{2}}{2g}}

where:

Φ represents the stability parameter, which adjusts the formula for different construction types:

φ = 1.25 at the end of a single layer of rock
φ = 1.0 at the end of bed protection
φ = 0.75 for two-layer rock bed protection
φ = 0.50 for continuous bed protections
the most conservative value is 1.5.

Ψ denotes the Shields parameter, with typical values being 0.035 for dumped rock and 0.05 for placed rock.

K_t = turbulence factor

k_t = 0.67 for low-turbulence flow
k_t = 1.0 for normal turbulence
k_t = 1.5 in river bends
k_t = 2.0 - 2.5 in sharp river bends (radius < 5 times river width)
k_t = 3.0 for high turbulence flow (e.g., propeller wash)
k_t = 4.0 for extreme turbulence (e.g., propeller wash at moored ships)

K_h = depth factor

Typically, $K_{h}={\frac {2}{\log({\frac {12h}{Nd}})^{2}}}$ , where N ranges from 1 to 3
k_h for an undeveloped velocity profile: $(1+{\frac {h}{d}})^{0.2}$

K_s = slope factor

The value of K_s corresponds to either $K(\alpha _{\parallel })$ or $K(\alpha )$ (as detailed below).

The destabilizing influences on a slope's stability can be quantified by examining two principal forces:

A component of force parallel to the slope, expressed as W sin(α), where W is the weight of the object and α is the slope angle. This force promotes downslope movement.
A component of force perpendicular to the slope, W cos(α), enhancing the normal force and thus the friction opposing movement.

In the figure below, φ represents the angle of internal friction or the angle of repose of the soil. When the flow direction aligns with the slope's inclination (Figure b), the perpendicular force impacting stability is:

F(\alpha _{\text{perpendicular}})=W\cos(\alpha )\tan(\phi )-W\sin(\alpha )

If the flow is in the opposite direction, the stone's stability increases:

F=W\cos(\alpha )\tan(\phi )+W\sin(\alpha )

The strength reduction factor due to the slope is then:

K(\alpha _{\parallel })={\frac {F(\alpha _{\parallel })}{F(0)}}={\frac {W\cos(\alpha )\tan(\phi )-W\sin(\alpha )}{W\tan(\phi )}}={\frac {\sin(\phi -\alpha )}{\sin(\phi )}}

For slopes transverse to the flow at an angle α (Figure c), the stability reduction factor is:

K(\alpha )={\frac {F(\alpha )}{F(0)}}={\sqrt {\frac {\cos ^{2}(\alpha )\tan ^{2}(\phi )-\sin ^{2}(\alpha )}{\tan ^{2}(\phi )}}}={\sqrt {1-{\frac {\sin ^{2}(\alpha )}{\sin ^{2}(\phi )}}}}

Figure d illustrates the reduction factors for stability at a slope with an angle of φ = 40 degrees, demonstrating the impact of slope angle relative to flow direction on the stability of objects.

Due to the fact that a depth factor, K_h, is included in this version of the Izbash formula, the average velocity above the stones can be considered for the velocity used in the calculations. This is a revision from the original Izbash formula, which ambiguously specified that the speed was "near to the stone".^[4]

Effect of turbulence

Relative velocity in a vortex near a stone Vortex-Hofland1.jpg — Relative velocity in a vortex near a stone

Turbulence exerts a significant effect on stability. Turbulent vortices cause locally high velocities at the stone, generating a lift force on one side, while the absence of such a force on the other side can eject the stone from its bed. This mechanism is depicted in the accompanying image (see right), where the detailed drawing illustrates the stone positioned at the coordinate (0,0), with the relative velocity creating an upward lift force to the left, and no lift force to the right, resulting in a clockwise moment that can flip the stone out of the bed along with the normal lift force of the main flow. This selective motion explains why not all stones are set in motion by a given current speed, but only when an appropriate vortex passes by.

Detail of the velocity near a stone Vortex-Hofland2.jpg — Detail of the velocity near a stone

These illustrations represent flow rate measurements in a vertical plane above a stone, with the flow moving from left to right. Displayed is the velocity after subtracting the average speed, i.e., the u and v components (for further explanation, see the main article on Turbulence modelling).

The impact of turbulence is particularly pronounced when the size of turbulent vortices is comparable to that of the stones. It is feasible to modify the Izbash formula to more explicitly incorporate the effects of turbulence.^[6] A stone at rest will not move until the total velocity (i.e., the average velocity plus the additional velocity from turbulent vortices) surpasses a specific threshold. Research indicates that this critical velocity is ${\overline {u}}+3\sigma =(1+3r){\overline {u}}$ , where $\sigma$ represents the standard deviation of the velocity and r denotes the relative turbulence. In the original Izbash formula, the coefficient of 1.7 encompasses a component accounting for turbulence. The formula can be reformulated as:

\Delta d=0.7{\frac {u^{2}}{2g}}=c_{iz}{\frac {\left[u(1+3r)\right]^{2}}{2g}}

Assuming a relative turbulence of r=0.075 for turbulence induced by bed roughness, the revised formula leads to $c_{iz}$ = 0.47. This revision introduces an explicit turbulence parameter into the Izbash formula:

\Delta d=0.47{\frac {\left[u(1+3r)\right]^{2}}{2g}}

This adaptation allows the use of the Izbash formula in scenarios where turbulence is not solely the result of bed roughness but also occurs in flows influenced by ships and propellers. In the wake of a vessel in a narrow channel, a strong return flow with increased turbulence is observed, where r is typically around 0.2. For propeller-induced turbulence, an r value of 0.45 is recommended.^[7] Given that the relative turbulence appears quadratically in the formula, it is evident that for a propeller flow, a substantially larger stone size is required for bed protection compared to "normal" flow conditions.

For non-typical cases, a turbulence model such as the k-epsilon model can be utilised to calculate the value of r. This value can then be inserted into the aforementioned modified Izbash formula to ascertain the necessary stone size.^[8]

Related Research Articles

In fluid dynamics, potential flow or irrotational flow refers to a description of a fluid flow with no vorticity in it. Such a description typically arises in the limit of vanishing viscosity, i.e., for an inviscid fluid and with no vorticity present in the flow.

<span class="mw-page-title-main">Angular velocity</span> Pseudovector representing an objects change in orientation with respect to time

In physics, angular velocity, also known as angular frequency vector, is a pseudovector representation of how the angular position or orientation of an object changes with time, i.e. how quickly an object rotates around an axis of rotation and how fast the axis itself changes direction.

The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields. This density and flux of energy and momentum are the sources of the gravitational field in the Einstein field equations of general relativity, just as mass density is the source of such a field in Newtonian gravity.

In fluid dynamics, Stokes' law is an empirical law for the frictional force – also called drag force – exerted on spherical objects with very small Reynolds numbers in a viscous fluid. It was derived by George Gabriel Stokes in 1851 by solving the Stokes flow limit for small Reynolds numbers of the Navier–Stokes equations.

Flight dynamics is the science of air vehicle orientation and control in three dimensions. The three critical flight dynamics parameters are the angles of rotation in three dimensions about the vehicle's center of gravity (cg), known as pitch, roll and yaw. These are collectively known as aircraft attitude, often principally relative to the atmospheric frame in normal flight, but also relative to terrain during takeoff or landing, or when operating at low elevation. The concept of attitude is not specific to fixed-wing aircraft, but also extends to rotary aircraft such as helicopters, and dirigibles, where the flight dynamics involved in establishing and controlling attitude are entirely different.

<span class="mw-page-title-main">Projectile motion</span> Motion of launched objects due to gravity

Projectile motion is a form of motion experienced by an object or particle that is projected in a gravitational field, such as from Earth's surface, and moves along a curved path under the action of gravity only. In the particular case of projectile motion on Earth, most calculations assume the effects of air resistance are passive and negligible. The curved path of objects in projectile motion was shown by Galileo to be a parabola, but may also be a straight line in the special case when it is thrown directly upward or downward. The study of such motions is called ballistics, and such a trajectory is a ballistic trajectory. The only force of mathematical significance that is actively exerted on the object is gravity, which acts downward, thus imparting to the object a downward acceleration towards the Earth’s center of mass. Because of the object's inertia, no external force is needed to maintain the horizontal velocity component of the object's motion. Taking other forces into account, such as aerodynamic drag or internal propulsion, requires additional analysis. A ballistic missile is a missile only guided during the relatively brief initial powered phase of flight, and whose remaining course is governed by the laws of classical mechanics.

In probability theory, a distribution is said to be stable if a linear combination of two independent random variables with this distribution has the same distribution, up to location and scale parameters. A random variable is said to be stable if its distribution is stable. The stable distribution family is also sometimes referred to as the Lévy alpha-stable distribution, after Paul Lévy, the first mathematician to have studied it.

An axial compressor is a gas compressor that can continuously pressurize gases. It is a rotating, airfoil-based compressor in which the gas or working fluid principally flows parallel to the axis of rotation, or axially. This differs from other rotating compressors such as centrifugal compressor, axi-centrifugal compressors and mixed-flow compressors where the fluid flow will include a "radial component" through the compressor.

The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. The theorem is a generalization of the second fundamental theorem of calculus to any curve in a plane or space rather than just the real line.

In geometry, various formalisms exist to express a rotation in three dimensions as a mathematical transformation. In physics, this concept is applied to classical mechanics where rotational kinematics is the science of quantitative description of a purely rotational motion. The orientation of an object at a given instant is described with the same tools, as it is defined as an imaginary rotation from a reference placement in space, rather than an actually observed rotation from a previous placement in space.

There are several equivalent ways for defining trigonometric functions, and the proofs of the trigonometric identities between them depend on the chosen definition. The oldest and most elementary definitions are based on the geometry of right triangles. The proofs given in this article use these definitions, and thus apply to non-negative angles not greater than a right angle. For greater and negative angles, see Trigonometric functions.

Sediment transport is the movement of solid particles (sediment), typically due to a combination of gravity acting on the sediment, and the movement of the fluid in which the sediment is entrained. Sediment transport occurs in natural systems where the particles are clastic rocks, mud, or clay; the fluid is air, water, or ice; and the force of gravity acts to move the particles along the sloping surface on which they are resting. Sediment transport due to fluid motion occurs in rivers, oceans, lakes, seas, and other bodies of water due to currents and tides. Transport is also caused by glaciers as they flow, and on terrestrial surfaces under the influence of wind. Sediment transport due only to gravity can occur on sloping surfaces in general, including hillslopes, scarps, cliffs, and the continental shelf—continental slope boundary.

The lateral earth pressure is the pressure that soil exerts in the horizontal direction. It is important because it affects the consolidation behavior and strength of the soil and because it is considered in the design of geotechnical engineering structures such as retaining walls, basements, tunnels, deep foundations and braced excavations.

The Orr–Sommerfeld equation, in fluid dynamics, is an eigenvalue equation describing the linear two-dimensional modes of disturbance to a viscous parallel flow. The solution to the Navier–Stokes equations for a parallel, laminar flow can become unstable if certain conditions on the flow are satisfied, and the Orr–Sommerfeld equation determines precisely what the conditions for hydrodynamic stability are.

Acoustic streaming is a steady flow in a fluid driven by the absorption of high amplitude acoustic oscillations. This phenomenon can be observed near sound emitters, or in the standing waves within a Kundt's tube. Acoustic streaming was explained first by Lord Rayleigh in 1884. It is the less-known opposite of sound generation by a flow.

The capstan equation or belt friction equation, also known as Euler-Eytelwein formula, relates the hold-force to the load-force if a flexible line is wound around a cylinder.

Blade element momentum theory is a theory that combines both blade element theory and momentum theory. It is used to calculate the local forces on a propeller or wind-turbine blade. Blade element theory is combined with momentum theory to alleviate some of the difficulties in calculating the induced velocities at the rotor.

The Shields formula is a formula for the stability calculation of granular material in running water.

<span class="mw-page-title-main">Van der Meer formula</span> Formula to calculate the stability of armourstone under wave action

The Van der Meer formula is a formula for calculating the required stone weight for armourstone under the influence of (wind) waves. This is necessary for the design of breakwaters and shoreline protection. Around 1985 it was found that the Hudson formula in use at that time had considerable limitations. That is why the Dutch government agency Rijkswaterstaat commissioned Deltares to start research for a more complete formula. This research, conducted by Jentsje van der Meer, resulted in the Van der Meer formula in 1988, as described in his dissertation. This formula reads

References

General reference: CIRIA, CUR, CETMEF (2007). The rock manual : the use of rock in hydraulic engineering. London: CIRIA C683. p. 1268. ISBN 9780860176831.{{cite book}}: CS1 maint: multiple names: authors list (link)

Footnotes

1 2 Izbash, Sergey Vladimirovich (1935). Construction of Dams by Dumping Stones into Flowing Water: (original title: Постройка плотин наброской камня в текущую воду) (in Russian). Leningrad: Scientific Research Institute of Hydrotechnics. p. 138.
1 2 Pilarczyk, Krystian (1998). Dikes and revetments, chapter 16. Delft, Netherlands: Balkema. ISBN 9789054104551.
↑ van den Berg, M. (2019). "Stability of randomly placed log bed protections". Masters Thesis. Retrieved 3 November 2023– via TU Delft Repository.
↑ Verhagen, H.J.; Mertens, M. (2010), "Riprap stability for deep water, shallow water and steep foreshores", Coasts, marine structures and breakwaters: Adapting to change, London: Thomas Telford Ltd, pp. 486–495, retrieved 3 November 2023
1 2 Hofland, Bas (2005). Rock & Roll, Turbulence-induced damage to granular bed protections. Delft: TU Delft. p. 241. ISBN 9789090201221 . Retrieved 22 March 2024.
↑ Schiereck, Gerrit Jan; Verhagen, Henk Jan (2012). Introduction to bed, bank and shoreline protection. Delft: Delft Academic Press. pp. 70–75. ISBN 978-90-6562-306-5. Archived from the original on 2019-04-13. Retrieved 22 March 2024.
↑ Verheij, H.J. (1988). Aanpassing van dwarsprofielen van vaarwegen (in Dutch). Delft: Delft Hydraulics. p. M1115/XIX.
↑ Watanabe, H. (1982). "Comment on Izbash's equation". Journal of Hydrology. 58 (3): 389–397. doi:10.1016/0022-1694(82)90048-8. ISSN 0022-1694 . Retrieved 3 November 2023.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[Izbash1935-1] 1 2 Izbash, Sergey Vladimirovich (1935). Construction of Dams by Dumping Stones into Flowing Water: (original title: Постройка плотин наброской камня в текущую воду) (in Russian). Leningrad: Scientific Research Institute of Hydrotechnics. p. 138.

[:0-2] 1 2 Pilarczyk, Krystian (1998). Dikes and revetments, chapter 16. Delft, Netherlands: Balkema. ISBN 9789054104551.

[3] van den Berg, M. (2019). "Stability of randomly placed log bed protections". Masters Thesis. Retrieved 3 November 2023– via TU Delft Repository.

[4] Verhagen, H.J.; Mertens, M. (2010), "Riprap stability for deep water, shallow water and steep foreshores", Coasts, marine structures and breakwaters: Adapting to change, London: Thomas Telford Ltd, pp. 486–495, retrieved 3 November 2023

[hofland-5] 1 2 Hofland, Bas (2005). Rock & Roll, Turbulence-induced damage to granular bed protections. Delft: TU Delft. p. 241. ISBN 9789090201221 . Retrieved 22 March 2024.

[Schiereck-6] Schiereck, Gerrit Jan; Verhagen, Henk Jan (2012). Introduction to bed, bank and shoreline protection. Delft: Delft Academic Press. pp. 70–75. ISBN 978-90-6562-306-5. Archived from the original on 2019-04-13. Retrieved 22 March 2024.

[M1115-XIX-7] Verheij, H.J. (1988). Aanpassing van dwarsprofielen van vaarwegen (in Dutch). Delft: Delft Hydraulics. p. M1115/XIX.

[8] Watanabe, H. (1982). "Comment on Izbash's equation". Journal of Hydrology. 58 (3): 389–397. doi:10.1016/0022-1694(82)90048-8. ISSN 0022-1694 . Retrieved 3 November 2023.

[1]

[2]

[3]

[4]

[6]

[7]

[8]