Sound Abnormally Stimulates the Vestibular System in Semicircular Canal Dehiscence Syndrome by Generating Pathological Fluid-Mechanical Waves

Sound Abnormally Stimulates the Vestibular System in Semicircular Canal Dehiscence Syndrome by Generating Pathological Fluid-Mechanical Waves

Why do some people get dizzy when hearing certain sounds? Researchers at University of Utah and Johns Hopkins1 have discovered why certain people experience dizziness when they hear a particular sound, such as a musical tone. For patients with (Superior) Semicircular Canal Dehiscence Syndrome (SSCD or SCDS),2, 3 there is a pathological hole in the bone that the inner ear is encased in, and certain acoustic tones cause the inner ear fluid to pump. As a result, the peripheral vestibular system portion of the cochlea sends an incorrect signal to the brain, causing dizziness and vertigo. We introduce the Science Report published in full4 in the journal Nature; and we add an engineering explanation of how Finite Element Analysis (FEA)5A, 5B, 44D is applied to the turbulent non-compressible fluid flow computational fluid dynamics (CFD)6 defined by the Navier-Stokes Equations7, as implemented in software such as COMSOL Multiphysics.9

In addition, while vetting references, we ran across Dr. Timothy Hain‘s latest update on his finding Ocular Vestibular Evoked Myogenic Potential (oVEMP)8 testing is apparently the most sensitive laboratory method of diagnosing SCDS, in his excellent Dizziness-and-Balance.com website; and we caution cochlear implant (CI) candidates — as well as surgeons — to radiologically rule out any latent SCDS so as to avoid “upsetting the applecart” when shoving an inch-long electrode into the pea-sized hearing & balance organ. We now present the Abstract and introduction; and then our extensive References and (especially!) “Bootnotes” on the vestibular-cochlear non-compressible fluid flow mechanics.

UPDATE: About an hour after we published this article, we received this message from a Facebook friend with adult-onset ANSD who we were guiding to get her CI’s:
Good timing… I was just reading your article that you posted and was getting ready to look up some information. I have experienced dizziness due to certain high pitch sounds. An example would be my dogs high pitch yippity bark. But on the CI, I was just working on that…
Abstract:

Individuals suffering from Tullio phenomena experience dizziness, vertigo, and reflexive eye movements (nystagmus) when exposed to seemingly benign acoustic stimuli. The most common cause is a defect in the bone enclosing the vestibular semicircular canals of the inner ear. Surgical repair often corrects the problem, but the precise mechanisms underlying Tullio phenomenon are not known. In the present work we quantified the phenomenon in an animal model of the condition by recording fluid motion in the semicircular canals and neural activity evoked by auditory-frequency stimulation. Results demonstrate short-latency phase-locked afferent neural responses, slowly developing sustained changes in neural discharge rate, and nonlinear fluid pumping in the affected semicircular canal. Experimental data compare favorably to predictions of a nonlinear computational model. Results identify the biophysical origin of Tullio phenomenon in pathological sound-evoked fluid-mechanical waves in the inner ear. Sound energy entering the inner ear at the oval window excites fluid motion at the location of the defect, giving rise to traveling waves that subsequently excite mechano-electrical transduction in the vestibular sensory organs by vibration and nonlinear fluid pumping.

Introduction:

Babylonian tablets scribed in the second millennium BC describe symptoms of vertigo, nystagmus, nausea and loss of balance – often disabling conditions attributed at the time to demonic possession rather than biology.10, 11 It was not until the Greek Hippocratic Corpus that consideration moved from demons to derangements of normal physiology to explain neuropsychiatric phenomena, clearing the path to establish a scientific understanding of balance disorders. The first key discoveries were made in the mid 19th century when Pierre Flourens and Prosper Ménière identified the inner ear semicircular canals (SCC) as the sensory organs responsible for angular motion sensation, and Josef Breuer identified the vestibulo-ocular reflex (VOR) as responsible for compensatory eye movements that stabilize the visual image on the retina by counteracting head movements.12,13 Normally, SCC afferent neurons exclusively encode and transmit angular head motion information to the brain, but become pathologically sensitive to linear acceleration, vibration, atmospheric pressure, and airborne sound if the temporal bone encasing the vestibular labyrinth is compromised by a fistula or dehiscence. SCC vestibular sensitivity to sound is referred to as Tullio phenomenon, named after Pietro Tullio who discovered that creating a fistula in the bony labyrinth leads to pathological SCC vestibular responses to sound.12,13 Patients suffering from Tullio phenomenon experience severe symptoms of sound-induced vertigo and ocular nystagmus. Lloyd Minor and colleagues14 identified dehiscence of the superior canal bony labyrinth as the most common cause, which has led to successful methods for diagnosis and surgical repair.15 But precisely why a fistula or dehiscence of the bony enclosure leads to Tullio phenomenon has remained a mystery for millennia.

Tullio phenomenon is characterized by sound-evoked nystagmus, with the eyes beating primarily in the plane of the affected canal.16,17,18,19 Sound-evoked eye movements are similar to those evoked in normal subjects by continuous angular acceleration of the head, demonstrating that sound evokes tonic semicircular canal responses in these subjects. Recordings from SCC afferent neurons after generating a small fistula in the bony labyrinth have revealed two characteristic types of pathological neural responses to pure tones: (1) neurons that lock action potential timing to a specific phase of the sinusoidal sound wave (phase-locking), and (2) neurons that increase or decrease action potential discharge rate during the sound stimulus without phase-locking (rate encoding).20 Phase-locking occurs primarily in neurons that fire action potentials with irregular inter-spike intervals, while rate encoding primarily occurs in neurons that that fire action potentials with regular inter-spike intervals.20,21,22 The low-frequency VOR relies primarily on inputs from regularly discharging afferent neurons driving the “sustained” vestibular system,23 while the high-frequency phasic VOR also relies on irregular phase-locking afferent neural inputs driving the “transient” system.24,25 Understanding how sound evokes inappropriate sustained and phase-locked vestibular inputs to the brain is therefore essential to understanding eye movements and origins of Tullio phenomenon.

A fistula or dehiscence is thought to give rise to Tullio phenomenon by introducing a flexible window in the bony labyrinth that diverts sound energy away from the cochlea and toward the affected canal.26,27.The temporal bone encasing the inner ear normally has only two flexible windows, both located in the middle ear. The oval window transmits sound from the middle ear stapes to the cochlea, while the round window is the pressure relief point. Introduction of a flexible “third window” diverts acoustic energy away from the cochlea and round window toward the affected semicircular canal, and this energy shunt explains hearing loss for air-conducted sounds caused by the condition.28,29The third mobile window also relieves pressure in the perilymph at the point of the fistula, thereby leading to a transmembrane pressure difference between endolymph and perilymph that can deform the membranous labyrinth, producing flow of endolymph that deflects sensory hair bundles in the SCC crista. This mechanism likely contributes to pressure and low-frequency infrasound sensitivity,30,31 but cannot account for sustained responses of afferent neurons to auditory frequency sound, nor can it account for frequency dependence of the magnitude and direction of eye movements. It has been suggested in previous studies of Tullio phenomena that nonlinear wave-driven fluid streaming32 or Liebau impedance pumping33,34,35 might underlie sustained responses to sound. The work of Grieser et al36 presents strong theoretical arguments that Tullio phenomena likely has origins in wave mechanics and nonlinear endolymph pumping in the deformable labyrinth. But sound-evoked neural responses measured in animal models and eye movements measured in humans show frequency dependent changes in excitation vs. inhibition that that are not described by current theories.

In the present work we combine theory and experiment to examine the hypothesis that traveling waves in the vestibular labyrinth give rise to phase-locked afferent neural responses by vibrating sensory hair bundles cycle-by-cycle, and give rise to sustained changes in afferent discharge rate by frequency dependent pumping of endolymph. Computational modeling of the human labyrinth37 was performed to elucidate biomechanics of the phenomena, and results were used to design specific experiments to confirm biophysics of the phenomena in an animal model. Experiments were performed in the oyster toadfish, Opsanus tau, an animal model selected to facilitate in vivo recording of afferent neurons and endolymph flow,38 and to provide reasonable morphological similarity to human.39

Experimental data shows steady endolymph pumping in the semicircular canals evoked by auditory frequency stimuli, as well as sustained afferent neuron responses to sound. The direction and magnitude were both frequency dependent, exhibiting multiple peaks and valleys in the auditory frequency spectrum. Results are consistent with mathematical analysis of nonlinear canal biomechanics and explain the origin of both sustained and transient vestibular responses in subjects suffering from Tullio phenomena.

Results:

To gain insight into the potential mechanical origin of Tullio phenomenon we used the finite element method (FEM) to simulate fluid motion and tissue deformation in the vestibular labyrinth. We constructed a simple FEM model based on the geometry of the human superior canal (Fig. 1A, SC) with a dehiscence (D) in the bone located half way around the loop. Endolymph and perilymph were modeled using the nonlinear Navier-Stokes equations and the membranous duct was modeled as a linear elastic tube. The morphology was simplified to an endolymph-filled elastic tube inside a rigid perilymph-filled bony tube (Fig. 1A,i). A pure tone acoustic pressure (1 A i, black arrows) was applied in the perilymph near the oval window (1 A. OW). Simulations predict the presence of traveling waves (TW), propagating away from the location of dehiscence and toward the location of acoustic stimulation Po (See Supplemental Video 1). Waves always traveled away from the dehiscence site and toward the pressure stimulus. This reverse propagation occurs because conservation of mass in the bony labyrinth forces the fluid displacement to be much larger near the small dehiscence relative to fluid displacement in the larger vestibule. Nearly identical results were obtained in our finite element model by applying a prescribed volume displacement at the site of the dehiscence and allowing pressure relief in perilymph at the round window., Hence, because the fluid is essentially incompressible, the mechanics can be examined using a volume velocity stimulus at the round window or an equivalent volume velocity at the dehiscence. This reciprocity motivated our animal model configuration using auditory frequency indentation in the membranous duct to induce endolymph volume velocity at the site of a simulated dehiscence.

Experimental design. (A) Surface reconstruction of an adult human bony labyrinth based on CT images illustrating the oval window (OW) where sound enters the inner ear, round window (RW), lateral canal (LC), posterior canal (PC), superior canal (SC) and location of dehiscence (D)59. (i) Navier-Stokes simulation with simplified toroidal canal geometry where sinusoidal pressure Po is applied at the outer tube (black arrows) and pressure is relieved at the dehiscence (D).40 The bony labyrinth was modeled as a perilymph-filled rigid tube (black outlines), and the membranous labyrinth was modeled as an endolymph-filled elastic tube. Color bar indicates the displacement magnitude of the membranous duct (white is zero, black is maximum). The maximum displacement occurs at the location of the dehiscence (D) and waves propagate away from the dehiscence towards the oval window stimulus site. (B) Membranous labyrinth of the experimental animal model (oyster toadfish), showing the location of the simulated dehiscence (SD) in the lateral canal (LC) and location of single-unit afferent neuron recordings in the LC nerve branch (E) (Adapted in part from Iversen et al41 with the permission of the Acoustical Society of America). (ii) The inverse of the inter-spike-interval, Spk-s-1 was recorded. (iii) Close-up of the lateral canal ampulla at the location where velocity fields were measured using PIV.

Based on FEM simulations44 we hypothesized that traveling waves give rise to phase-locked afferent responses by vibrating sensory hair bundles as the waves pass through the ampulla, and give rise to sustained afferent responses by sustained pumping of endolymph in the direction of wave propagation. Cycle-by-cycle vibration of hair bundles is clearly supported by the simple FEM model (Fig. 1A,i), but in order for sustained pumping to occur in a preferred direction (e.g. counter clockwise, ampullofugal vs. clockwise, ampullopetal) there must be some asymmetry in the morphology that allows one of the two traveling waves to dominate. To investigate this idea we used the morphologically-descriptive 1-D model of Iversen and Rabbitt42 to analyze endolymph pumping. In Fig. 2 the spatial distribution of the sustained component of endolymph pressure caused by nonlinear fluid pumping is shown as a color map and the vibrational component of the transmembrane pressure is displayed in the polar plot. Like the FEM model, waves are predicted by the 1-D model to travel away from the site of simulated dehiscence (Fig. 2) toward the vestibule (See Supplemental Videos 2–3). However, due to the asymmetric geometry, traveling waves on one side of the dehiscence dominate, leading to sustained endolymph pumping in a frequency-dependent preferred direction. The reason why one side can dominate is illustrated in Fig. 2C (790 Hz, lower panel), where the ampullopetal wave reflects as it travels toward the utricle leading to standing waves (SW) while the ampullofugal wave propagates with less reflection. This causes net fluid pumping in the ampullofugal direction. Local reflection of traveling waves is caused by variations in the acoustic-wave impedance introduced by local changes in membranous duct cross-sectional area. The stiffness of the cupula is unimportant relative to fluid mass, fluid viscosity, and membranous duct elasticity for auditory frequency stimuli.42 Since the wavelength is frequency dependent and the vibrational patterns are frequency dependent,42 the direction and magnitude of nonlinear endolymph pumping also depend on frequency.

1-D Mechanical simulations. Endolymph pressure and transmembrane pressure in a human lateral canal with simulated dehiscence in response to auditory-frequency stimulation at 419 Hz (top) and 790 Hz (bottom) shown at three instances in time (0, τ, 3τ s, where τ in this simulation was 11.8 s). The spatial distribution of the sustained component of endolymph pressure (Equation 4pe1) caused by nonlinear fluid pumping is shown as a color map (black is minimum, yellow is maximum) and the vibrational component of transmembrane pressure (p0 = pe0 − pp0) as a polar plot (black solid line relative to gray dotted line). Waves originate from the location of dehiscence and propagate towards the vestibule leading to cycle-by-cycle vibration of sensory hair bundles in the ampulla. Black arrows indicate direction of net fluid displacement, q, predicted to be opposite at 419 vs. 790 Hz. (A) At the beginning of the stimulus, waves propagate away from the dehiscence with zero initial pressure gradient across the cupula. (B) Over time, waves pump endolymph in a frequency-dependent preferred direction leading to a pressure gradient across the cupula and tonic deflection of sensory hair bundles in the ampulla. (C) At 3τ s, the pressure gradient across the cupula reaches maximum while waves continue to propagate away from the dehiscence and vibrate the tissue. At 790 Hz (C, lower), transmembrane pressure waves are shown at two instants in time to illustrate traveling waves (TW) on one side of the dehiscence and standing waves (SW) on the other.

If the theoretical predictions in Figs 1 and 2 underlie Tullio phenomena in the living ear, we hypothesized it would be possible to generate both phase-locked afferent responses and frequency-dependent sustained responses in a simple animal model. To test this hypothesis, we recorded single-unit responses of afferent neurons and endolymph fluid velocity evoked by auditory frequency stimuli. The oyster toadfish lateral canal (LC) with a simulated dehiscence (Fig. 1B, SD) was used to facilitate in vivo neural recordings and particle imaging velocimetry (PIV).43 As expected from simulations, afferent neurons exhibited both phase-locked and sustained responses, depending on the specific neuron and frequency tested. Four examples of phase-locked responses are shown in Fig. 3 for stimuli at 422, 500 and 800 Hz. Notice that these afferents had a rapid onset consistent with the rapid wave propagation from the site of the simulated dehiscence to the ampulla. Phase-locked units also exhibited some adaptation during the stimulus, followed by slow recovery after the stimulus, both consistent with slow endolymph pumping superimposed on the vibration. Consistent with previous reports in this species, SCC afferents that phase-locked to auditory frequency stimuli had irregular baseline inter-spike-intervals, as quantified for this specific population by their high coefficient of variation of 0.40 +/− 0.19.22 These units phase-locked to the stimulus with high vector strength 0.85 +/− 0.11 and winding ratios between 2–5.

Please see the article for the balance of the the Results, plus the Discussion, Methods, Modeling Methods, and More References. Also, please note they use the surface integral version of the Navier-Stokes Equations; while our explanation uses, what we believe, the more intuitive partial differential version.

References:
  1. University of Utah. “Why some people get dizzy when hearing certain sounds.” ScienceDaily. (accessed July 16, 2018);
  2. Superior Canal Dehiscence — New: oVEMP testing is most sensitive laboratory method of diagnosing SCD. Timothy C. Hain, MD; Marcello Cherchi, M.D. • Page last modified: June 27, 2018
  3. Superior Semicircular Canal Dehiscence Syndrome – Diagnosis and Surgical Management [Entire article].  International Archives of Otorhinolaryngology, 2017 Apr; 21(2): 195–198. Marite Palma DiazJuan Carlos Cisneros Lesser, and Alfredo Vega Alarcón. doi:  10.1055/s-0037-1599785; PMCID: PMC5375705; PMID: 28382131.
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  5. A. What is FEA | Finite Element Analysis?  SimScale SimWiki;  B. Finite Element Method – What Is It? FEM and FEA Explained   By: Ajay Harish.  Updated on March 22nd, 2018
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  19. Ostrowski, V. B., Byskosh, A. & Hain, Timothy C. Tullio phenomenon with dehiscence of the superior semicircular canal. Otol Neurotol 22, 61–65 (2001).
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  22. Curthoys, I. S., MacDougall, H. G., Vidal, P. P. & de Waele, C. Sustained and Transient Vestibular Systems: A Physiological Basis for Interpreting Vestibular FunctionFront Neurol 8, 117,  (2017);
  23. Iversen, M. M. et al. Low-intensity ultrasound activates vestibular otolith organs through acoustic radiation forceJ Acoust Soc Am141, 4209,  (2017);
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  25. Minor, L. B., Lasker, D. M., Backous, D. D. & Hullar, T. E. Horizontal vestibuloocular reflex evoked by high-acceleration rotations in the squirrel monkey. I. Normal responses. J Neurophysiol 82, 1254–1270 (1999);
  26. Huizinga, E. On the sound reaction of Tullio in the pigeon and the fenestration operation in man. Acta Otolaryngol Suppl 100, 174–180 (1952);
  27. Songer, J. E. & Rosowski, J. J. A mechano-acoustic model of the effect of superior canal dehiscence on hearing in chinchillaJ Acoust Soc Am 122, 943–951, (2007);
  28. Songer, J. E. & Rosowski, J. J. A superior semicircular canal dehiscence-induced air-bone gap in chinchillaHear Res 269, 70–80,  (2010);
  29. Rabbitt, R., Boyle, R. & Highstein, S. Influence of surgical plugging on horizontal semicircular canal mechanics and afferent response dynamicsJ Neurophysiol 82(2), 1033–1053 (1999);
  30. Grieser, B. J., Kleiser, L. & Obrist, D. Identifying Mechanisms Behind the Tullio Phenomenon: a Computational Study Based on First PrinciplesJ Assoc Res Otolaryngol 17, 103–118, (2016);
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  34. Iversen, M. & Rabbitt, R. Wave mechanics of the vestibular semicircular canalsBiophysical J 113, 1–17,  (2017);
  35. Rabbitt, R. D., Boyle, R. & Highstein, S. M. Mechanical amplification by hair cells in the semicircular canalsProc Natl Acad Sci USA 107, 3864–3869,  (2010);
  36. Ghanem, T. A., Rabbitt, R. D. & Tresco, P. A. Three-dimensional reconstruction of the membranous vestibular labyrinth in the toadfish, Opsanus tauHear Res 124, 27–43 (1998);
  37. Straka, H., Zwergal, A. & Cullen, K. E. Vestibular animal models: contributions to understanding physiology and diseaseJ Neurol263(Suppl 1), S10–23,  (2016);
  38. Rabbitt, R. D., Boyle, R., Holstein, G. R. & Highstein, S. M. Hair-cell versus afferent adaptation in the semicircular canalsJ Neurophysiol 93, 424–436,  2005).
  39. Curthoys, I. S., Vulovic, V., Sokolic, L., Pogson, J. & Burgess, A. M. Irregular primary otolith afferents from the guinea pig utricular and saccular maculae respond to both bone conducted vibration and to air conducted soundBrain Res Bull 89, 16–21,  (2012).
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  41. Deutschlander, A., Jahn, K., Strupp, M. & Brandt, T. Pulse-synchronous rotational and vertical pendular eye movements in superior canal dehiscence syndromeEuropean journal of neurology 14, e29, 2007).
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Bootnotes:

2) The Navier-Stokes Equations, as explained by the rocket scientists at NASA’s Glenn Research Center in Cleveland and Plum Brook Station annex in nearby Sandusky, Ohio [We edited the HTML to replace the original partial differential symbol partialgraphic with the correct “∂” unicode character, for proper search engine indexing ~DLS]:

In this figure, we show the three-dimensional unsteady form of the Navier-Stokes Equations. These equations describe how the velocity, pressuretemperature, and density of a moving fluid are related. The equations were derived independently by Sir George Stokes, in England, and Claude-Louis Navier, in France, in the early 1800’s. The equations are extensions of the Euler Equations and include the effects of viscosity on the flow.

Previously, in Viscosity Vs Hardness Of Ear Impression Materials, we presented the definition of viscosity:

Brookfield viscometers and various assorted spindles

Brookfield viscometers and various assorted spindles. Click to enlarge

• Viscosity is a measure of a fluid’s resistance to flow, describing the internal friction of it while moving. For example, imagine a styrofoam cup with a hole in the bottom. If we then pour honey into the cup we will find that the cup drains very slowly. That is because honey’s viscosity is large compared to other liquids’ viscosities. If we fill the same cup with water, the cup will drain much more quickly.

The SI physical unit of viscosity is the Poise, which is in pascal-seconds (Pa•s), which is equivalent to N•s/m² or kg/(m•s) — in the lab it is measured with a Brookfield viscometer, which measures the resistance of a spinning disc or drum — is a measure of a fluid’s resistance to flow, describing the internal friction of it while moving. For example, imagine a Styrofoam cup with a hole in the bottom. If we then pour honey into the cup we will find that the cup drains very slowly. That is because honey’s viscosity is large compared to other liquids’ viscosities. If we fill the same cup with water, the cup will drain much more quickly.

Returning to the NASA presentation:

These equations are very complex, yet undergraduate engineering students are taught how to derive them in a process very similar to the derivation that we present on the conservation of momentum web page.

The equations are a set of coupled differential equations and could, in theory, be solved for a given flow problem by using methods from calculus. But, in practice, these equations are too difficult to solve analytically. In the past, engineers made further approximations and simplifications to the equation set until they had a group of equations that they could solve. Recently, high speed computers have been used to solve approximations to the equations using a variety of techniques like finite difference, finite volume, finite element, and spectral methods. This area of study is called Computational Fluid Dynamics or CFD.

The Navier-Stokes equations consists of a time-dependent continuity equation for conservation of mass, three time-dependent conservation of momentum equations and a time-dependent conservation of energy equation. There are four independent variables in the problem, the xy, and z spatial coordinates of some domain, and the time t. There are six dependent variables; the pressure p, density r, and temperature T (which is contained in the energy equation through the total energy Et) and three components of the velocity vector; the u component is in the x direction, the v component is in the y direction, and the w component is in the z direction, All of the dependent variables are functions of all four independent variables. The differential equations are therefore partial differential equations and not the ordinary differential equations that you study in a beginning calculus class.

You will notice that the differential symbol is different than the usual “d /dt” or “d /dx” that you see for ordinary differential equations. The symbol “∂” is is used to indicate partial derivatives. The symbol indicates that we are to hold all of the independent variables fixed, except the variable next to symbol, when computing a derivative. The set of equations are:

Continuity: ∂r/∂t + ∂(r * u)/∂x + ∂(r * v)/∂y + ∂(r * w)/∂z = 0X – Momentum: ∂(r * u)/∂t + ∂(r * u2)/∂x + ∂(r * u * v)/∂y + ∂(r * u * w)/∂z = – ∂p/∂x + 1/Re * { ∂tauxx/∂x + ∂tauxy/∂y + ∂tauxz/∂z}

Y – Momentum: ∂(r * v)/∂t + ∂(r * u * v)/∂x + ∂(r * v2)/∂y + ∂(r * v * w)/∂z = – ∂p/∂y + 1/Re * { ∂tauxy/∂x + ∂tauyy/∂y + partialtauyz/∂z}

Z – Momentum: ∂(r * w)/∂t + ∂(r * u * w)/∂x + ∂(r * v * w)/∂y + ∂(r * 2)/∂z = – ∂p/∂z + 1/Re * { ∂tauxz/∂x + ∂tauyz/∂y + ∂tauzz/∂z}

Energy: ∂Et/∂t + ∂(u * Et)/∂x + ∂(v * Et)/∂y + ∂(w * Et)/∂z = – ∂(r * u)/∂x – ∂(r * v)/∂y – ∂(r * w)/∂z  – 1/(Re*Pr) * { ∂qx/∂x + ∂qy/∂y + ∂qz/∂z} + 1/Re * {∂(u * tauxx + v * tauxy + w * tauxz)/∂x + ∂(u * tauxy + v * tauyy + w * tauyz)/∂y + ∂(u * tauxz + v * tauyz + w * tauzz)/∂z}

…where Re is the Reynolds number

Here, we’ll provide a more formal definition of what the Reynolds number Re is than NASA provides, and how it applies to non-compressible fluids, such as endolymph, while NASA presents it the context of compressible fluids, such as aerodynamics and hot exhaust gas flow.

What is the Reynolds Number?

The dimensionless Reynolds number plays a prominent role in foreseeing the patterns in a fluid’s behavior. The Reynolds number, referred to as Re, is used to determine whether the fluid flow is laminar or turbulent. It is one of the main controlling parameters in all viscous flows where a numerical model is selected according to pre-calculated Reynolds number.

Although the Reynolds number comprises both static and kinematic properties of fluids, it is specified as a flow property since dynamic conditions are investigated. Technically speaking, the Reynolds number is the ratio of the inertial forces and the viscous forces. In practice, the Reynolds number is used to predict if the flow will be laminar or turbulent.

If the inertial forces, which resist a change in velocity of an object and are the cause of the fluid movement, are dominant, the flow is turbulent. Otherwise, if the viscous forces, defined as the resistance to flow, are dominant – the flow is laminar. The Reynolds number can be specified as below:

(1)

 

For instance, a glass of water which stands on a static surface, regardless of any forces apart from gravity, is at rest and flow properties are ignored. Thus, the numerator in equation (1) is “0”. That results in independence from the Reynolds number for a fluid at rest. On the other hand, whilst water is spilled by tilting a water-filled glass, flow properties abide by physical laws, a Reynolds number can be estimated as to predict fluid flow that is illustrated in Figure 1.

Figure 1: A glass of water which is a) at rest; b) flows.

Derivation

The dimensionless Reynolds number predicts whether the fluid flow would be laminar or turbulent referring to several properties such as velocity, length, viscosity, and also type of flow. It is expressed as the ratio of inertial forces to viscous forces and can be explained in terms of units and parameters respectively, as below:

where  is the density of the fluid,  is the characteristic velocity of the flow, and L(m) is the characteristic length scale of flow. (4) Equation (3) is the derivation of units at which the Reynolds number is specified as non-dimensional. Variations of the Reynolds number are shown in equation (2) where  is the dynamic viscosity of fluid and  is the kinematic viscosity. The transition between dynamic and kinematic viscosity is as follows:

Fluid, Flow and Reynolds Number

The applicability of the Reynolds number differs depending on the specifications of the fluid flow such as the variation of density (compressibility), variation of viscosity (Non-Newtonian), being internal- or external flow etc. The critical Reynolds number is the expression of the value to specify transition among regimes which diversifies regarding type of flow and geometry as well. Whilst the critical Reynolds number for turbulent flow in a pipe is 2000, the critical Reynolds number for turbulent flow over a flat plate, when the flow velocity is the free-stream velocity, is in a range from 10 ⁵ to 10⁶.

The Reynolds number also predicts the viscous behavior of the flow in case fluids are Newtonian. Therefore, it is highly important to perceive the physical case to avoid inaccurate predictions. Transition regimes and internal as well as external flows with either low or high Reynolds number in use, are the basic fields to comprehensively investigate the Reynolds number. Newtonian fluids are fluids that have a constant viscosity. If the temperature stays the same, it does not matter how much stress is applied on a Newtonian fluid; it will always have the same viscosity. Examples include water, alcohol and mineral oil.

Laminar to turbulent transition

The fluid flow can be specified under two different regimes: Laminar and Turbulent. The transition among the regimes is an important issue that is driven by both fluid and flow properties. As mentioned before, the critical Reynolds number, which changes in accordance with the physical case, can be classified as internal and external, where it might face slight changes in the amount. Yet while the Reynolds number regarding the laminar-turbulent transition can be defined reasonably for internal flow, it is hard to specify a definition for an external flow.

Laminar flow (top), turbulent flow (bottom)

 

Laminar flow: For practical purposes, if the Reynolds number is less than 2000, the flow is laminar. The accepted transition Reynolds number for flow in a circular pipe is Red,crit = 2300. Laminar flow properties include:

  • Re < 2000
  • ‘Low’ velocity
  • Fluid particles move in straight lines
  • Layers of water flow over one another at different speeds with virtually no mixing between layers.
  • The flow velocity profile for laminar flow in circular pipes is parabolic in shape, with a maximum flow in the center of the pipe and a minimum flow at the pipe walls.
  • The average flow velocity is approximately one half of the maximum velocity.
  • Simple mathematical analysis is possible.

Turbulent flow (top) vs laminar flow (bottom). Note the scale on the left indication the transition when 2000 ≤ Re ≤ 4000

Turbulent flow:

  • Re > 4000
  • ‘high’ velocity
  • The flow is characterized by the irregular movement of particles of the fluid.
  • Average motion is in the direction of the flow
  • The flow velocity profile for turbulent flow is fairly flat across the center section of a pipe and drops rapidly extremely close to the walls.
  • The average flow velocity is approximately equal to the velocity at the center of the pipe.
  • Mathematical analysis is very difficult.
  • Most common type of flow, and is used in the Navier-Stokes equations for cochlear fluid flow modeling.

In addition, there is a third class of flows, Transitional flow: At Reynolds numbers 2000 ≤ Re ≤ 4000 the flow is unstable as a result of the onset of turbulence. These flows are sometimes referred to as transitional flows; and are in the light blue band in the second photo.

Returning to the NASA presentation:

The q variables are the heat flux components and Pr is the Prandtl number which is a similarity parameter that is the ratio of the viscous stresses to the thermal stresses. The tau variables are components of the stress tensor. A tensor is generated when you multiply two vectors in a certain way. Our velocity vector has three components; the stress tensor has nine components. Each component of the stress tensor is itself a second derivative of the velocity components.

The terms on the left hand side of the momentum equations are called the convection terms of the equations. Convection is a physical process that occurs in a flow of gas in which some property is transported by the ordered motion of the flow. The terms on the right hand side of the momentum equations that are multiplied by the inverse Reynolds number are called the diffusion termsDiffusion is a physical process that occurs in a flow of gas in which some property is transported by the random motion of the molecules of the gas. Diffusion is related to the stress tensor and to the viscosity of the gas. Turbulence, and the generation of boundary layers, are the result of diffusion in the flow. The Euler equations contain only the convection terms of the Navier-Stokes equations and can not, therefore, model boundary layers. There is a special simplification of the Navier-Stokes equations that describe boundary layer flows.

Notice that all of the dependent variables appear in each equation. To solve a flow problem, you have to solve all five equations simultaneously; that is why we call this a coupled system of equations. There are actually some other equation that are required to solve this system. We only show five equations for six unknowns. An equation of state relates the pressure, temperature, and density of the gas. And we need to specify all of the terms of the stress tensor. In CFD the stress tensor terms are often approximated by a turbulence model.

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About the author

Dan Schwartz

Electrical Engineer, via Georgia Tech

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